backward stochastic differential equations with jumps are ... › bitstream › 11303 › ... ·...

Backward Stochastic Differential Equations

with Jumps are Stable

vorgelegt von

Diplom Mathematiker

Alexandros Saplaouras

geb. in Athen

Von der Fakultat II - Mathematik und Naturwissenschaften

der Technischen Universitat Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

Dr. rer. nat

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. Fredi TroltzschGutachter: Prof. Dr. Antonis PapapantoleonGutachter: Prof. Dr. Peter FrizGutachter: Prof. Dr. Dylan Possamaı

Tag der wissenschaftlichen Aussprache: 18.07.2017

Berlin 2017

Στους αγαπημένους μου γονείς και στα αδέρφια μου

Contents

Abstract iZusammenfassung iiAcknowledgements iiiSynopsis ivBasic Notation and Terminology v

Chapter I. Elements of Stochastic Calculus 1I.1. Preliminaries and notation 2I.2. Uniform integrability 7I.3. Uniformly integrable martingales 16I.4. Stochastic integration 18I.5. Selected results from stochastic calculus 26I.6. The Skorokhod space 29I.7. Martingale representation of square-integrable martingales 46

Chapter II. Backward Stochastic Differential Equations with Jumps 51II.1. Introduction 51II.2. A simple counterexample and its analysis 52II.3. Suitable spaces and associated results 54II.4. Framework and statement of the theorem 57II.5. A priori estimates 60II.6. Proof of the theorem 68II.7. Related literature 71

Chapter III. Stability of Martingale Representations 73III.1. Framework and statement of the main theorem 73III.2. Outline of the proof of the main theorem 75III.3. To be (orthogonal), or not to be, here is the answer 77III.4. Constructing a sufficiently rich family of convergent martingale-sequences 81III.5. The orthogonal martingale-sequence 90III.6. Θ∞, I is an F–martingale, for I ∈ J (X∞). 91III.7. Step 1 is valid 101III.8. Corollaries of the main theorem 102III.9. Comparison with the literature 103

Chapter IV. Stability of Backward Stochastic Differential Equations with Jumps 105IV.1. Notation 105IV.2. Framework and statement of the main theorem 107IV.3. Proof of Theorem IV.7 109IV.4. The first step of the induction is valid 113IV.5. The p−step of the induction is valid 117IV.6. Comparison with the related literature 131

Appendix A. Auxiliary results 133A.1. Auxiliary results of Chapter II 133A.2. Auxiliary results of Chapter III 138A.3. Auxiliary results of Chapter IV 141

Appendix. Bibliography 149

i

Abstract

A backward stochastic differential equation is a stochastic differential equation whose terminal valueis known, in contrast to a (forward) stochastic differential equation whose initial value is known, andwhose solution has to be adapted to a given filtration. The main aim of this thesis is to provide thesuitable framework for the stability of stochastic differential equations with jumps, hereinafter BSDEsor BSDE when we refer to a single object. With the term stability we understand the continuity of theoperator that maps the standard data of a BSDE, a set which among others includes the terminal valueof the BSDE and the filtration with respect to which the solution has to be adapted, to its solution. Inother words, the stability property allows to obtain an approximation of the solution of the BSDE underinterest, once we determine an approximation of the standard data of the BSDE under interest.

In this thesis we provide a general wellposedness result of multidimensional BSDEs with stochasticLipschitz generator and which is driven by a possibly stochastically discontinuous square-integrable mar-tingale. The time horizon can be infinite and as already implicitly has been stated, the right-continuousfiltration is allowed to be stochastically discontinuous. Moreover, we provide a framework under whichthe stability property of BSDEs is verified. This framework allows for both continuous-time and discrete-time L2−type approximations, which can turn out to be particularly useful for the well-posedness ofnumerical schemes for BSDEs. These results are presented in the second and the fourth chapter of thisthesis. In the third chapter the stability of martingale representations is obtained, a result which lies atthe core of the stability property of BSDEs. The property of the stability of martingale representationsis not only a useful tool for our current needs, but it is also an interesting result on its own. Roughlyspeaking, it amounts to the convergence of the spaces generated by a convergent sequence of stochasticintegrators as well as of their corresponding orthogonal spaces.

Apart from these main results, a series of other results have been obtained, which either improveor complement classical ones. The most interesting of them is of purely analytic nature. It provides acharacterisation of the weak-convergence of finite measures on the positive real-line by means of relativelycompact sets of the Skorokhod space endowed with the J1−topology. We remain in the Skorokhod space,where we refine a classical result on convergence of the jump-times of a J1−convergent sequence. Moreprecisely, we deal with the case of a multidimensional J1−convergent sequence and we prove that thetimes that the heights of the jumps lie in a suitable fixed set form a convergent sequence in the extendedpositive real-line. We proceed with the theory of Young functions, where the contribution amounts to thefollowing result. We prove that the conjugate Young function of the composition of a moderate Youngfunction with R+ 3 x 7→ 1

2x2 ∈ R+ is also a moderate Young function with further nice properties.

Finally, a new inequality regarding generalised inverses complements a classical one.

Zusammenfassung

Eine ruckwarts-stochastische Differentialgleichung (BSDE) ist eine stochastische Differentialgleichung,dessen Endwert gegeben ist. Dies steht im Gegensatz zu vorwarts-stochastischen Differentialgleichungen,bei denen der Anfangswert gegeben ist und die Losung an eine Filtration angepasst werden muss. Zieldieser Arbeit ist, die mathematischen Grundlagen fur die Stabilitat von stochastischen Differentialgle-ichungen mit Sprungen zu formulieren. Unter Stabilitat verstehen wir hier die Stetigkeit des Operators,welcher die Daten der BSDE – diese beinhalten u.a. den Endwert und die entsprechende Filtration – aufihre Losung abbildet. In anderen Worten ermoglicht die Stabilitatseigenschaft, eine Approximation derLosung der BSDE zu erhalten, sobald wir eine Approximation der Daten der BSDE haben.

Die vorliegende Arbeit beinhaltet ein allgemeines Resultat zur Wohlgestelltheit von mehrdimension-alen BSEDs mit stochastischem Lipschitz Generator, die von einem moglicherweise stochastischen undunstetigen aber integrierbaren Martingal gesteuert wird. Wir erlauben einen unendlichen Zeithorizontsowie stochastische Unstetigkeiten in der rechtsstetigen Filtration. Daruber hinaus analysieren wir Bedin-gungen unter der die Stabilitatseigenschaft der BSDE erfullt ist. Dies erlaubt sowohl zeitkontinuierlicheals auch zeitdiskrete L2−Approximationen, welche speziell fur die Wohl-gestelltheit numerischer Ver-fahren fur BSDEs von Nutzen sein kann. Die entsprechenden Resultate befinden sich im zweiten undvierten Kapitel dieser Arbeit. Im dritten Kapitel wird die Stabilitat der Martingal Darstellungen gezeigt.Diese Eigenschaft liefert nicht nur ein fur diese Arbeit wichtiges Hilfsmittel sondern stellt an sich schonein interessantes Ergebnis in Hinblick auf die Konvergenz von Raumen, die durch konvergente Folgen vonstochastischen Integratoren generiert werden, dar.

Neben den erwahnten Hauptresultaten liefert diese Arbeit noch eine Reihe weiterer Ergebnisse. Dasvielleicht Interessanteste ist rein algebraisch und liefert eine Charakterisierung der schwachen Konver-genz endlicher Maße auf der positiven reellen Achse relativ kompakter Mengen des Skorokhod Raumsausgestattet mit der J1−Topologie. Zudem gibt es eine Verfeinerung eines klassischen Resultats uber dieSprungzeiten von J1−konvergenten Folgen. Schließlich wird noch bewiesen, dass die konjugierte YoungFunktion der Komposition einer moderaten Young Funktion mit R+ 3 x 7→ 1

2x2 ∈ R+ eine moderate

Young Funktion mit guten Eingeschaften ist.

Acknowledgements

A few words, sometimes, are not enough to express oneself their gratitude to somebody. However,a few typed words are enough to communicate to a random reader this feeling. The next lines willbe served as such to express my gratitude to my supervisor, Antonis Papapantoleon or more preciselyΑντώνη Παπαπαντολέοντα, for the uncountable hours he has devoted for mathematical discussions andespecially those comments which improved my understanding and intuition in the purely-discontinuousworld. Moreover, I would like to thank him for the trust he has put in me in many different ways. Σεευχαριστώ για όλα, Αντώνη!

I would like to thank Peter K. Friz and Dylan Possamaı for agreeing to referee my thesis. I regardtheir presence in the committee not only a great honour, but also a great responsibility in order to meettheir high standards. I would like to express especially to Dylan my thanks for the warm hospitality Ihave received in my many visits in Paris.

Next I would like to thank Peter Bank for his pieces of advice and his useful comments, which did notlie only around mathematical issues, and Peter Imkeller for co-supervising my thesis. Moreover, I wouldlike to thank Samuel Cohen for pointing out a mistake in the proof of the theorem regarding the stabilityproperty of martingale representations. Our discussion has greatly inspired me to finally discover theright path.

I am grateful to the DGF Research Training Group 1845 “Stochastic Analysis with Applications inBiology, Finance and Physics”, not only for the financial support I have received, but also for the chanceto be a member of its family. I have enjoyed every single time. I would like to explicitly mention Todor,who was always interested in discussing. His comments were much appreciated.

I would like to thank many more people for different reasons, but I will only mention their names asa thank you. Ευχαριστώ Παναγιώτα, Robert, Χρήστο, Νατάσα, Δάντη-Ευάγγελε, ΄Ολγα, Αλεξάνδρα, Eva,. . .

Finally, I gratefully acknowledge the partial financial support from the IKYDA program 54718970“Stochastic analysis in finance and physics” and the PROCOPE project “Financial markets in transition:mathematical models and challenges”.

Synopsis

The dissertation, which is based on joint work with Antonis Papapantoleon and Dylan Possamaı, isdivided in four chapters. At the beginning of each chapter, except for the first one, we provide someintroductory comments about the topic of the chapter, while at the end of each one, again the exceptionis the first chapter, we compare our results with the existing literature. In the first chapter we providemost of the notation as well as the machinery we are going to use. The experienced reader may skimthrough it in order to identify the notation, which in most of the cases is the one commonly used inCohen and Elliott [20], Dellacherie and Meyer [26, 27], Jacod and Shiryaev [41] and He, Wang, and Yan[35]. We have tried to make the dissertation self-contained, so even the non-experienced reader may havedirect access to the arguments we use. In the remaining results of this chapter either we provide sufficientreference for their proof or we provide the proof when we want to underline specific points.

However, in the first chapter there are also results which are new, to the best knowledge of the author.These are

• The right inequality in Lemma I.34.(vii), which turns out to be powerful enough for obtaining the apriori estimates in Chapter II, see Lemma II.16.• Proposition I.51 which provides some nice properties of moderate Young functions, which we believe

may prove fruitful for convergence results under an L2−setting.• Subsection I.6.1, where we prove that the times the jump of a J1−convergent sequence form a

| · |−convergent sequence. This result will be used in Chapter III in order to construct a family ofJ1−convergent sequences of submartingales.

Moreover, Section I.7 can be regarded as a new contribution in the sense that it provides a new perspective.To be more precise, it is not the difficulty of the results we present in this section, but that we set thesuitable framework under which we obtain a slight generalisation of the classical result [41, LemmaIII.4.24]. This in turn, will permit the co-existence of an Ito stochastic integrator with jumps and aninteger-valued random measure, which is in particular the case in numerical schemes.

In Chapter II we provide a general wellposedness result for multidimensional BSDEs with possiblyunbounded random time horizon and driven by a general martingale in a filtration only assumed tosatisfy the usual hypotheses, i.e. the filtration may be stochastically discontinuous. We show that forstochastic Lipschitz generators and unbounded, possibly infinite, time horizon, these equations admit aunique solution in appropriately weighted spaces. Our result allows in particular to obtain a wellposednessresult for BSDEs driven by discrete-time approximations of general martingales. The current chapterrelies heavily on [54] but in virtue of Section I.7 we obtain a generalisation which allows the integratorof the Ito stochastic integral to be a square-integrable martingale with non-trivial purely discontinuouspart, as has been described above.

In Chapter III the core of the thesis is presented, the stability of martingale representations. In thefirst section we settle the framework and we state the main theorem. Then we present the preparatoryresults we need for the proof, which is finally presented in Section III.7.

Finally, in Chapter IV we present the result that justifies the title of the dissertation. Here we followa different approach for presenting the proof. We provide the main arguments in Section IV.3 in orderto reduce the complexity of the problem under consideration. Then, in the remaining sections of thischapter we prove that the pair of sufficient conditions we have determined are indeed satisfied.

Basic Notation and Terminology

∅ denotes the empty set

R denotes the set of real numbers

[a, b) = x ∈ R, a ≤ x < b, similarly for [a, b], (a, b], (a, b)

|x| denotes the absolute value of the real number x

R+ = [0,∞)

R+ = [0,∞]

N = 1, 2, 3, . . . , i.e. the set of positive integers

p, q denote positive integers. More generally, pi, qi denote positive integers, for i in an arbitrary index-set

` is a fixed positive integer

N = N ∪ ∞Rp = the Euclidean p−dimensional space

(Rp×q,+) is the group of p× q−matrices with entries in R(yi)i∈I denotes a family indexed by the set I

x ∧ y = min(x, y), for x, y ∈ RAc denotes the complement of the set A

1A denotes the indicator function of the set A

δα denotes the Dirac measure sitting at α

a.s. is the abbreviation for “almost surely”

σ(A) denotes the σ−algebra generated by the class AC ⊗D denotes the product σ−algebra of the classes C and DB(E) denotes the Borel σ−algebra of the metric space E

s ↑ t stands for s→ t with s < t. Analogously, s ↓ t stands for s→ t with s > t

f(t−), lims↑t

f(s) := lims→ts<t

f(s), i.e. the left-limit of f at the point t ∈ (0,∞), if it exists

f(t+), lims↓t

f(s) := lims→ts>t

f(s), i.e. the right-limit of f at the point t ∈ R+, if it exists∫A

f(z)µ(dz) denotes the Lebesgue–Stieltjes integral of f with respect to the measure µ over the set A∫A

f(z) dz denotes the Lebesgue–Stieltjes integral of f with respect to the Lebesgue measure over the set A

CHAPTER I

Elements of Stochastic Calculus

In this chapter we outline the notation as well as the definitions and the results from stochasticcalculus we will make use of. The main references will be the books of Cohen and Elliott [20], ofDellacherie and Meyer [26, 27], of He et al. [35] and of Jacod and Shiryaev [41]. For [26, 27] we will usethe following convention: we will refer to the Statement number 2 of Chapter I by Statement I.2, p. 7,i.e [26, Definition I.2, p. 7]1 refers to the definition of a random variable. For [41], when we want to referto the Statement number 3 in Section 1 of Chapter I, we will write Statement I.1.3, i.e. [41, DefinitionI.1.3] refers to the definition of the complete stochastic basis.

The basic notation has been provided on page v. However, before we proceed to the first sectionof the chapter we provide additional notation which completes the basic one. Since we deal with limittheorems with possibly multi-dimensional objects and in order to familiarise the reader with the notationwe use, we will reserve some letters for specific purposes. More specifically, the letters k, l and m willserve as indexes of the elements of sequences and they will be assumed to lie in N. The letters i, j, p andq are reserved to denote arbitrary natural integers. In the case of p, q, the same holds true when theyare indexed by a natural integer, e.g. p37 ∈ N. The aforementioned letters will represent (mainly) thedimension of the spaces or the position of the elements of a multi-dimensional object. The calligraphicletter ` will be assumed to be a fixed positive integer, except for Chapter IV where we assume ` = 1. It isgoing to denote exclusively the dimension of the martingales which will serve as “stochastic integrators”.The capital letter E, whether it is indexed (e.g. E1) or not, will denote either an Euclidean space or thegroup (Rp×q,+). Therefore (E,+) is always a group. We abuse notation and denote by 0 the neutralelement of the (arbitrary) group (E,+).

Let us fix a matrix v ∈ Rp×q,2 its transpose will be denoted by v>∈ Rq×p. The element at the i−throw and j−th column of v will be denoted by vij , for 1 ≤ i ≤ p and 1 ≤ j ≤ q and it will be called the(i, j)−element of v. However, the notation needs some care when we deal with sequences of elements ofRp×q, e.g. if (vk)k∈N ⊂ Rp×q then we will denote by vk,ij the (i, j)−element of vk, for every 1 ≤ i ≤ p,

1 ≤ j ≤ q and for every k ∈ N. We will identify Rp with R1×p, i.e. the arbitrary x ∈ Rp will be identifiedas a row-vector of length p. The i−th element of x ∈ Rp will be denoted by xi, for 1 ≤ i ≤ p. Moreover,if (xk)k∈N ⊂ Rp then xk,i denotes the i−th element of xk, for every 1 ≤ i ≤ p and for every k ∈ N. The

trace of a square matrix w ∈ Rp×p is given by Tr[w] :=∑pi=1 w

ii. We endow the space E := Rp×q withthe norm ‖ · ‖2, defined by ‖v‖22 = Tr[v>v] and we remind the reader that this norm is derived from theinner product defined for any (v, w) ∈ Rp×q × Rp×q by Tr[v>w]. We will additionally endow E with thenorm ‖ · ‖1, which is defined by ‖v‖1 :=

∑pi=1

∑qj=1 |vij |. The associated to E Borel σ−algebra will be

denoted by B(E). The space E is always finite-dimensional, therefore no confusion arises with respect towhich topology the Borel σ−algebra is meant, since the norms ‖·‖1 and ‖·‖2 generate the same topology.Finally, let (F, δ) be a complete metric space. If the sequence (xk)k∈N ⊂ F converges to x∞ ∈ F underthe metric δ, then we will say that sequence (xk)k∈N is a δ−convergent sequence and we will denote it

by xkδ−−−−→

k→∞x∞3. If the space F is a (real) vector space and is endowed with a norm ‖ · ‖, then the

metric induced by the norm ‖ · ‖ will be denoted by δ‖·‖. In this case we will say that a sequence is‖ · ‖−convergent or δ‖·‖−convergent interchangeably.

1In [26] the page numbering consists of the number of the page and the letter of the chapter. In order to avoid therepetition of the chapter-letter we will write only the number of the page.

2Observe that we have not mentioned explicitly that p, q ∈ N. That was the purpose of the previous paragraph. Inorder to avoid repetitions, no further reference will be made for the aforementioned letters.

3We will omit the index associated to the convergence, i.e. k →∞, whenever it is clear to which index the convergencerefers.

1

2 I. ELEMENTS OF STOCHASTIC CALCULUS

The following notational abuse will considerably simplify some statements. For a given finite familyof spaces (Ei)i∈1,...,q we endow its Cartesian product with the ‖ · ‖1−norm

q∏i=1

Ei 3 e = (e1, . . . , eq)‖·‖17−−−−→

q∑i=1

‖ei‖1

as well as with the ‖ · ‖2−norm

q∏i=1

Ei 3 e = (e1, . . . , eq)‖·‖27−−−−→

( q∑i=1

‖ei‖22) 1

2

.

Let, now, f : R+ → Rp×q be a function. The value of the function f at point s will be denoted byf(s), while the value of its (i, j)−element at point s will be denoted by f ij(s). If p = 1 the value of itsi−element at point s will be denoted by f i(s). If f is real-valued, it will be said non-decreasing (resp. non-increasing , increasing , decreasing) if for every 0 ≤ s < t holds f(s) ≤ f(t) (resp. f(s) ≥ f(t),f(s) < f(t),f(s) > f(t)).

Let us fix a function f : (R+, δ|·|) → (E, δE). We define f(0−) := f(0), f(t−), lims↑t f(s) :=lims→t

s<tf(s) for t > 0 and f(t+), lims↓t f(s) := lims→t

t<sf(s), whenever the limits exist and are elements of

E. If for every t ∈ R+ the values f(t−), f(t+) are well-defined and f(t) = f(t+), resp. f(t−) = f(t),then the function will be called cadlag , resp. caglad . The French abbreviation cadlag stands for continua droite avec limite a gauche, i.e. right-continuous with left limits, while caglad stands for continu agauche avec limite a droite, i.e. left-continuous with right limits. If f is cadlag, then we associate to itthe caglad function f− : (R+, δ|·|)→ (E, δE) defined by f−(0) := f(0) and f−(t) := f(t−), for t ∈ (0,∞),as well as the function ∆f : (R+, δ|·|) → (E, δE) defined by ∆f := f − f−. If ∆f(t) 6= 0, then the pointt will be said to be a point of discontinuity for f . We close this paragraph with the following definition.If the limit lims→∞ f(s) is well-defined, then we extend the function f on R+ and we define the value off at the symbol ∞ to be f(∞) := lims→∞ f(s).

We conclude this part with the definition of some useful functions. The identity function on R, i.e.R 3 x 7−→ x ∈ R, will be denoted by Id, while the identity function on R` will be denoted by Id`. Thecanonical i−projection Rp 3 x 7−→ xi ∈ R will be denoted by πi, for 1 ≤ i ≤ p, where we suppress in

the notation the indication of the domain. Finally, we define the functions R+ 3 xquad7−−−→ 1

2x2 ∈ R+ and

R` 3 x q7−→ x>x ∈ R`×`.

I.1. Preliminaries and notation

A probability space (Ω,G,P) is a triplet consisting of the sample space Ω, of the σ−algebra G on Ω,whose elements are called events, and the probability measure P defined on the measurable space (Ω,G).Every probability space we are going to use will be assumed to be complete, i.e. the σ−algebra G iscomplete. The latter means that all P−negligible sets belong to G. The arbitrary subset of Ω× R+ willbe referred to as random set . A measurable function ξ : (Ω,G)→ (E,B(E)) will be called an (E−valued)random variable. A family M = (Mt)t∈R+ of E−valued random variables will be called (E−valued)stochastic process or simply (E−valued) process. When the process M is considered as a mapping fromthe measurable space (Ω×R+,G ⊗B(R+)) into (E,B(E)) we will say that M is G ⊗B(R+)−measurable.Mt denotes the value of the process at time t, while for fixed ω ∈ Ω the function R+ 3 t 7−→Mt(ω) ∈ Eis called the ω−path of the process M .

The arbitrary probability space (Ω,G,P) is fixed for the rest of the chapter.

For this section E = Rp×q. We will say that a property P holds P−almost surely, abbreviatedby P − a.s., if P(ω ∈ Ω, P (ω) is true) = 1. A random set A will be called evanescent if the setω ∈ Ω,∃t ∈ R+ with (ω, t) ∈ A is P−negligible. Observe that, since the probability space is complete,we are eligible for writing P(ω ∈ Ω,∃t ∈ R+ with (ω, t) ∈ A) = 0. Two processes M1,M2 will be calledindistinguishable if the random set ω ∈ Ω,∃t ∈ R+ with M1

t (ω) 6= M2t (ω) is evanescent.

We introduce some more notational conventions. For the rest of the section we will denote by ζa real-valued random variable and by ξ, ξ1, ξ2 etc. E−valued random variables. The property ξ1 ≤ ξ2stands for ξ1 ≤ ξ2 P−a.s. element-wise, i.e. the elements of ξ2−ξ1 are non-negative real numbers P−a.s.,with the obvious interpretation for ξ1 ≥ ξ2, ξ1 < ξ2, ξ1 > ξ2, ξ1 = ξ2 and ξ1 6= ξ2. Moreover, we define[ξ1 ≤ ξ2] := ω ∈ Ω, ξ1(ω) ≤ ξ2(ω) and [ξ1 ≥ ξ2], [ξ1 < ξ2], [ξ1 > ξ2], [ξ1 = ξ2], [ξ1 6= ξ2] analogously.

I.1. PRELIMINARIES AND NOTATION 3

Expectations under P will be denoted by E[·], i.e.

E[ζ] :=

∫Ω

ζ(ω) dP(ω) =

∫Ω

ζ dP or E[ξ] :=

∫Ω

ξ(ω) dP(ω) =

∫Ω

ξ dP,

where in the latter case the expectation is calculated element-wise, therefore E[ξ] is a p × q−matrix. IfE[|ζ|] < ∞ (resp. E

[‖ξ‖1

]< ∞, which is equivalent to E

[‖ξ‖2

]< ∞) then ζ (resp. ξ) will be called

integrable. For F a sub−σ−algebra of G we define the spaces

L1(F ;E) :=ξ is an F −measurable E−valued random variable,E

[‖ξ‖1

]<∞

and

L2(F ;E) :=ξ is an F −measurable E−valued random variable,E

[‖ξ‖22

]<∞

,

where P − a.s. equal random variables belong to the same class of equivalence. For ϑ ∈ 1, 2, thecorresponding norm ‖ · ‖Lϑ(F ;E) is defined by ‖ · ‖ϑLϑ(F ;E) := E

[‖ · ‖ϑϑ

]. The Riesz–Fischer Theorem, e.g.

see Cohen and Elliott [20, Theorem 1.5.33], let us know that the normed spaces (L1(F ;E), ‖ · ‖1) and(L2(F ;E), ‖ · ‖2) are Banach. Choose a ϑ ∈ 1, 2 and let (ξk)k∈N, (ηt)t∈R+

⊂ Lϑ(F ;E). If

E[‖ξk − ξ∞‖ϑϑ

]−−−−→k→∞

0, resp. E[‖ηt − η∞‖ϑϑ

]−−−→t↑∞

0,

then we will say interchangeably that the sequence (ξk)k∈N converges to ξ∞ in Lϑ(F ;E)−mean or that thesequence (ξk)k∈N is Lϑ(F ;E)−convergent, resp. the family (ηt)t∈R+

converges to η∞ in Lϑ(F ;E)−mean

or that the family (ηt)t∈R+is Lϑ(F ;E)−convergent . We will denote the above convergence by

ξkLϑ(F ;E)−−−−−−→ ξ∞, resp. ηt

Lϑ(F ;E)−−−−−−→ η∞.

The convergence in probability under a metric δ of the sequence (ηk)k∈N will be denoted by

ηk(δ,P)−−−−→ η∞.

The conditional expectation of ξ with respect to F will be denoted by the E−valued F−measurablerandom variable E[ξ|F ], where E[ξ|F ]ij := E[ξij |F ] for 1 ≤ i ≤ p, 1 ≤ j ≤ q. Since we do not requireany integrability property for the random variable ξ, we are implicitly making use of the generalisedconditional expectation; see He et al. [35, Section I.4] or Jacod and Shiryaev [41, Definition 1.1].

Definition I.1. An E−valued stochastic process M will be called cadlag (resp. continuous, caglad) if

P([R+ 3 t 7−→Mt ∈ E is cadlag (resp. continuous, caglad)]

)= 1.

When M is cadlag, we define the E−valued stochastic processes M− := (Mt−)t∈R+and ∆M :=

(∆Mt)t∈R+, where

M0−(ω) := M0(ω),Mt−(ω) := lims↑t

Ms(ω) and ∆Mt(ω) := Mt(ω)−Mt−(ω)

for every ω ∈ Ω such that lims↑tMs(ω) is well-defined.

Definition I.2. A family G = (Gt)t∈R+of sub−σ−algebrae of G will be called filtration if it is

(i) increasing, i.e. Gs ⊂ Gt for every 0 ≤ s ≤ t,(ii) right-continuous, i.e. Gt = ∩u>tGu for every t ∈ R+, and

(iii) complete, i.e. the σ−algebra G04 contains all P−negligible sets of G.

We abuse the usual convention and we set G∞,G∞− := σ(A ⊂ G,∃t ∈ R+ such that A ∈ Gt

).

The reader familiar with the standard terminology of stochastic calculus will recognise that a filtrationis defined in such a way that it satisfies the usual conditions a priori. This implies that whenever we makeuse of the natural filtration of M , which will be denoted by FM , we will refer to the usual augmentationof FM+ = (FMt+)t∈R+

, where Ft+ is defined by FMt+ :=⋂u>t σ

(Ms, s ≤ u

)for every t ∈ R+.

Definition I.3. The probability space (Ω,G,P) endowed with a filtration G = (Gt)t∈R+will be referred

to as the G−stochastic basis and it will be denoted by (Ω,G,G,P). When it is clear to which filtrationwe refer, we will simply write stochastic basis, i.e. we will omit the symbol of the filtration from thenotation.

The probability space (Ω,G,P) is endowed with an arbitrary filtration G for the rest of the chapter.

4In view of (i) the σ−algebra Gt contains all P−negligible sets of G, for every t > 0.


Definition I.4. Let M be an E−valued stochastic process.

(i) We will say that the process M is adapted to the filtration G if Mt is Gt−measurable for everyt ∈ R+. We will also use indifferently the term M is G−adapted .

(ii) An E−valued process M is a G−martingale (resp. G−submartingale, G−supermartingale) on theG−stochastic basis if Mt ∈ L1(Gt;E)5 for every t ∈ R+ and

E[Mt|Gs] = Ms (resp. E[Mt|Gs] ≥Ms, E[Mt|Gs] ≤Ms) for every 0 ≤ s ≤ t.

(iii) We will say that the E−valued process M is non-decreasing (resp. non-increasing , increasing ,decreasing) if M0 = 0 P − a.s. and its paths are cadlag and non-decreasing, (resp. non-increasing,increasing, decreasing) P−almost surely. In this case limt↑∞Mt exists P − a.s. and therefore we defineM∞(ω) := limt↑∞Mt(ω) for every ω for which the limit is well-defined.

(iv) We will say that the E−valued process M is of finite variation if M0 = 0 P−a.s. and its paths arecadlag with finite variation over each compact interval of R+ P−almost surely. In this case, the variation

process of M is also an E−valued process defined as Var(M)ijt (ω) := Var(M ij(ω))t, for every 1 ≤ i ≤ p,1 ≤ j ≤ q and denoted by Var(M) :=

(Var(M)t

)t∈R+

.

Definition I.5. Let X,Y be real-valued G−martingales.(i) If their product XY is a G−martingale, then X and Y will be called (mutually) G−orthogonal

and this will be denoted by X ⊥⊥ Y.(ii) The G−martingale X will be called purely discontinuous G−martingale if X ⊥⊥ M for every

continuous G−martingale M.

Theorem I.6. Any E−valued G−martingale X admits a unique, up to indistinguishability, decomposi-tion

X = X0 +Xc +Xd,

where Xc0 = Xd

0 = 0, Xc is a continuous G−martingale and Xd is a purely discontinuous G−martingale.

Proof. Apply Jacod and Shiryaev [41, Theorem I.4.18] element-wise.

Definition I.7. Let X be a G−martingale. The unique continuous G−martingale Xc, resp. purelydiscontinuous G−martingale Xd, associated to X by Theorem I.6 will be called the continuous part ofX , resp. the purely discontinuous part of X . The pair (Xc, Xd) will be called the natural pair of X underG.

Given the new terms introduced in the previous theorem, it is a good point to introduce some furthernotation.

Notation I.8. Let (Xk)k∈N be a sequence of Rp−valued martingales. Then, for k ∈ N, 1 ≤ i ≤ p• Xk,c,i denotes the i−element of the continuous part of Xk.• Xk,d,i denotes the i−element of the purely discontinuous part of Xk.

Corollary I.9. Let X and Y be two purely discontinuous G−martingales having the same jumps, i.e.∆X = ∆Y up to indistinguishability. Then X and Y are indistinguishable.

Proof. Apply Theorem I.6 to X − Y .

We proceed with the definition of two σ−algebrae on Ω× R+ which are of utmost importance.

Definition I.10. (i) The G−optional σ−algebra is the σ−algebra OG on Ω × R+ which is gener-ated by all cadlag and G−adapted processes considered as mappings on Ω × R+. A process which isOG−measurable will be called G−optional process.

(ii) The G−predictable σ−algebra is the σ−algebra PG on Ω × R+ which is generated by all left-continuous and G−adapted processes considered as mappings on Ω×R+. A PG−measurable process willbe called G−predictable process.

Notation I.11. Now we can provide part of the notation for the spaces we are going to use as well assome further convenient notation.

5Observe that the integrability condition implies that M is G−adapted.

I.1. PRELIMINARIES AND NOTATION 5

• M(G;E) will denote the space of all uniformly integrable G−martingales,6 i.e. the space of allG−martingales X such that the family (‖Xt‖1)t∈R+ is uniformly integrable; see Definition I.24 for thenotion of uniform integrability.• Mc(G;E) := X ∈M(G;E), X is continuous and• Md(G;E) := X ∈M(G;E), Xij is purely discontinuous for 1 ≤ i ≤ p, 1 ≤ j ≤ q.• H2(G;E) will denote the space of square integrable G−martingales, i.e. the space of G−martingales

X such that supt∈R+E[‖Xt‖22

]<∞.

• H2,c(G;E) := H2(G;E) ∩Mc(G;E).7

• H2,d(G;E) := H2(G;E) ∩Md(G;E).• V+(G;E) will denote the space of all E−valued, cadlag, G−adapted and non-decreasing processes.• V(G;E) will denote the space of all E−valued, cadlag, G−adapted and of finite variation processes.• V+

pred(G;E) := V ∈ V+, V is G−predictable• Vpred(G;E) := V ∈ V, V is G−predictable.• A+(G;E) := A ∈ V+, A∞ := limt↑∞At ∈ L1(G;E).• A(G;E) := A ∈ V,Var(A)∞ := limt↑∞Var(A)t ∈ L1(G;E).• A+

pred(G;E) := V+pred(G;E) ∩ A+(G;E).

• Apred(G;E) := Vpred(G;E) ∩ A(G;E).• Ssp(G;E) will denote the space of all G−adapted processes S which can be written in the form

S = S0 + X + A, where S0 is E-valued and G0−measurable, X ∈ M(G;E) and A ∈ Vpred(G;E). Theelements of Ssp(G;E) will be said G−special semimartingales.

• If R+ 3 tf7−→ f(t) ∈ E is a function, we will abuse notation and we will denote in the same way the

process Ω× R+ 3 (ω, t)f7−→ f(t) ∈ E.

• Given two filtrations F := (Ft)t∈R+, H := (Ht)t∈R+

, we will say that F is a sub-filtration of H if forevery t ∈ R+ holds Ft ⊂ Ht.

Definition I.12. The elements of the space Ssp(G;E) will be called G−special semimartingales. The(unique up to indistinguishability) decomposition S = S0 + X + A such that X ∈ M(G;E) and A ∈Vpred(G;E) is called the G−canonical decomposition of S.

Definition I.13. (i) The mapping τ : Ω → R+ is called G−stopping time if τ ≤ t ∈ Gt for everyt ∈ R+.

(ii) Let X be a G−optional process and τ be a G−stopping time. The process Xτ is defined byXτt := Xτ∧t and will be said to be the stopped process X at time τ .(iii) Let τ be a G−stopping time. We define Gτ :=

A ∈ G, A∩τ ≤ t ∈ Gt for all t ∈ R+

. Moreover,

Gτ− := σ(G0 ∪

A ∩ τ < t, t ∈ R+ and A ∈ Gt

).

(iv) Let τ1, τ2 be G−stopping times. We define the stochastic interval by

Kτ1, τ2K := (ω, t) ∈ Ω× R+, τ1(ω) < t ≤ τ2(ω)and analogously for Jτ1, τ2K, Jτ1, τ2J, Kτ1, τ2J. In particular, we will denote Jτ1, τ1K by Jτ1K and we will referto it as the graph of the stopping time τ1.

(v) Let ρ be a G−stopping time. If J0, ρJ∈ PG, then ρ will be called G−predictable (stopping) time.(vi) Let σ be a G−stopping time. If there exists a sequence (ρk)k∈N of G−predictable times such that

JσK ⊂ ∪k∈NJρkK, then σ will be called G−accessible (stopping) time.(vii) Let τ be a G−stopping time. If P([τ = ρ] ∩ [τ <∞]) = 0 for every G−predictable time ρ, then τ

will be called G−totally inaccessible (stopping) time.

Theorem I.14. The G−predictable σ−algebra is generated by any one of the following collections ofrandom sets:

(i) A× 0 for A ∈ G0, and J0, τK where τ is a G−stopping time.(ii) A× 0 for A ∈ G0, and A× (t, u] where t < u, A ∈ Gt.

Definition I.15. (i) The filtration G is said to be quasi-left-continuous if each G−accessible time isa G−predictable time.

6In view of Definition I.2 and Dellacherie and Meyer [27, Theorem VI.4, p. 69] for every element of M(G;E) we can

choose a cadlag and G−adapted modification. Therefore, we can assume, and we will do so, that all the elements of the

space M(G;E) are cadlag. The interested reader should consult [27, Section 1 of Chapter VI] for the general discussionregarding cadlag modifications of supermartingales.

7It is a consequence of Theorem I.27 that H2(G, ;E) ⊂M(G;E).


(ii) A cadlag G−adapted process M is called G−quasi-left-continuous if ∆Mρ1[ρ<∞] = 0 P− a.s. forevery G−predictable time ρ.

The following characterisations will be useful.

Theorem I.16. The filtration G is quasi-left-continuous if and only if every element of M(G;R) isG−quasi-left-continuous.

Proof. See He et al. [35, Theorem 5.36]. We inform the reader that our definition and [35, Definition3.39] differ, but [35, Theorem 3.40] proves that they are equivalent.

Definition I.17. A random set A is called thin if there exists a sequence of G−stopping times (τk)k∈Nsuch that A = ∪k∈NJτkK. If, moreover, the sequence (τk)k∈N satisfies JτmK∩ JτnK = ∅ for all m 6= n, thenit will be called an exhausting sequence for the thin set A.

Proposition I.18. If X is a cadlag and G−adapted process, the random set [∆X 6= 0] is thin. Anexhausting sequence for the set [∆X 6= 0] is called a sequence that exhausts the jumps of X.

Proof. See [41, Proposition I.1.32].

Proposition I.19. Let X be a cadlag and G−adapted process. Then, the following are equivalent:

(i) X is G−quasi-left-continuous.(ii) There exists an exhausting sequence of G−totally inaccessible stopping times that exhausts the

jumps of X.

Proof. See Jacod and Shiryaev [41, Proposition I.2.26].

We close this section with the following theorems.

Theorem I.20 (Existence of the G−optional projection). Let M be a G ⊗B(R+)−measurable E−valuedprocess such that for every G−stopping time τ holds E

[‖Xτ‖11[τ<∞]

]< ∞. Then there exists a unique

E−valued G−optional process, denoted by ΠGo (M), such that for every G−stopping time τ we have

E[Mτ1[τ<∞]|Gτ ] = ΠGo (M)τ1[τ<∞] P− a.s..

The process ΠGo (M) will be called the G−optional projection of M .

Proof. See He et al. [35, Theorem 5.1].

Theorem I.21 (Existence of the G−predictable projection). Let M be a G⊗B(R+)−measurable E−valuedprocess such that for every G−predictable time ρ holds E

[‖Xρ‖11[ρ<∞]

]<∞. Then there exists a unique

E−valued G−predictable process, denoted by ΠGp (M), such that for every G−predictable time ρ we have

E[Mρ1[ρ<∞]|Gρ] = ΠGp (M)ρ1[ρ<∞] P− a.s..

The process ΠGp (M) will be called the G−predictable projection of M .


Theorem I.22. Let M be a G ⊗ B(R+)−measurable E−valued process and X be a G−optional, resp.G−predictable process. If the G−optional projection, resp. G−predictable projection, of M exists, thenthe G−optional projection, resp. G−predictable projection, of XM exists. In this case,

ΠGo (XM) = X ΠG

o (M), resp. ΠGp (XM) = X ΠG

p (M).

The following result provides a convenient property of the optional projection of a process. Recallthat its wellposedness is verified by Theorem I.20. The reader may anticipate to the Subsection I.4.1 forthe Lebesgue–Stieltjes integral of a process A.

Lemma I.23. Let A ∈ V(G;R) and Y be a uniformly integrable and measurable process. Then for everyG−stopping time holds

E[ ∫

(0,τ ]

Ys dAs

]= E

[ ∫(0,τ ]

ΠGo (Y )s dAs

]. (I.1)

Proof. See Cohen and Elliott [20, Theorem 8.1.21].

I.2. UNIFORM INTEGRABILITY 7

I.2. Uniform integrability

We devote this section to the central concept of uniform integrability, which lies behind many im-portant results of stochastic analysis. As we are mainly interested in the limit behaviour of sequences ofmartingales, we need to mention that it is the uniform integrability of the sequence which guarantees thatthe limit will remain a martingale. The precise statement is given at Proposition I.147. This classicalresult is going to be one of the cornerstones in Chapter III. Now we proceed to present the requiredmachinery for proving that a family of random variables is uniformly integrable, whose main tool will bethe class of moderate Young functions.

Definition I.24. Let U ⊂ L1(G;R). The set U is said to be a uniformly integrable subset of L1(G;R) if

limc→∞

supξ∈U

E[|ξ|1[|ξ|>c]] = 0.

Theorem I.25. The set U ⊂ L1(G;R) is uniformly integrable if and only if the following conditions areboth satisfied

(i) supξ∈U E[|ξ|] <∞ and(ii) for any ε > 0, there exists a δ > 0 such that, supξ∈U E[|ξ|1A] < ε for all A ∈ G with P(A) < δ.

Proof. See Dellacherie and Meyer [26, Theorem II.19, p. 22] or Cohen and Elliott [20, Theorem2.5.4] or He et al. [35, Theorem 1.9].

Theorem I.26 (Dunford–Pettis Compactness Criterion). Let U ⊂ L1(G;R). Then the following areequivalent:

(i) U is uniformly integrable.(ii) U is relatively compact in L1(G;R) endowed with the weak8 topology.

(iii) U is relatively weakly sequentially compact, i.e. every sequence (ξk)k∈N ⊂ U contains a subsequence(ξkl)l∈N which converges weakly.

Proof. See Dellacherie and Meyer [26, Theorem II.25, p. 27].

Theorem I.27 (de La Vallee Poussin). Let U ⊂ L1(G;R). Then U is uniformly integrable if and only ifthere exists a function Υ : R+ → R+ such that

limx→∞

Υ(x)

x=∞ and sup

ξ∈UE[Υ(|ξ|)] <∞.

Proof. See Dellacherie and Meyer [26, Theorem II.22, p. 24].

Remark I.28. Let U ⊂ L1(G) be a uniformly integrable set and Υ be the function associated to U byTheorem I.27. In Subsection I.2.2, more precisely in Corollary I.48, we will see that we can choose thefunction Υ to be additionally convex and moderate; see Notation I.41 for the latter. We will refer toCorollary I.48 as the de La Vallee Poussin–Meyer criterion.

In the following corollary we collect some convenient results regarding uniform integrability.

Corollary I.29. (i) Let (ξi)i∈I ⊂ L1(G;R) be a uniformly integrable family and α ∈ R. Then (αξi)i∈Iis uniformly integrable.

(ii) Let (ξi)i∈I , (ζi)i∈I ⊂ L1(G;R) with |ζi| ≤ |ξi| for every i ∈ I. If (ξi)i∈I is uniformly integrable, thenso it is (ζi)i∈I .

(iii) Let ξ ∈ L1(G;R) and (Fi)i∈I be a family of sub−σ−algebrae of G. Then the family(E[ξ|Fi]

)i∈I is

uniformly integrable.(iv) Let (ξki )i∈I ⊂ L1(G;R) be uniformly integrable for k = 1, . . . , p. Then the family

(∑pk=1 ξ

ki

)i∈I is

uniformly integrable.(v) Let Ak := (ξki )i∈Ik ⊂ L1(G;R) be uniformly integrable for k = 1, . . . , p. Then the family ∪pk=1Ak

is uniformly integrable.

Proof. (i) It follows immediately by the definition.(ii) It follows immediately by the definition.(iii) See He et al. [35, Theorem 1.8].

8We use the probabilistic convention for the term weak. In functional analysis this is the weak?-topology, which is also

denoted by σ(L1,L∞).


(iv) It is immediate from Theorem I.25. However, we provide the proof for the convenience of thereader.

For the condition (i) of the aforementioned theorem we have

supi∈I

E[ ∣∣∣ p∑

k=1

ξki

∣∣∣ ] ≤ supi∈I

E[ p∑k=1

|ξki |]≤

p∑k=1

supi∈I

E[|ξki |]<∞,

where we have used that for every k ∈ 1, . . . , p the family (ξki )i∈I satisfies Theorem I.25, thereforesupi∈I E

[|ξki |]<∞.

We proceed to prove condition (ii) of Theorem I.25. To this end, let ε > 0. Then, for everyk ∈ 1, . . . , p, since the family (ξki )i∈I is uniformly integrable, there exists a δk(ε, p) > 0 such thatsupi∈I E

[|ξki |1A] < ε

p for all A ∈ G with P(A) < δk(ε, p). Define δ(ε, p) := minδk(ε, p), k = 1, . . . , p.Then it is immediate that

supi∈I

E[ ∣∣∣ p∑

k=1

ξki

∣∣∣1A] ≤ supi∈I

E[ p∑k=1

|ξki |1A]≤

p∑k=1

supi∈I

E[|ξki |1A

]<

p∑k=1

ε

p= ε.

(v) One can argue as in the proof of (iv) by choosing the smallest δ among the δ’s which are obtainedby Theorem I.25. Alternatively, by the Dunford–Pettis compactness criterion and using the classicalargument for proving that the finite union of sequentially compact sets is sequentially compact.

The following definition provides the analogous of Definition I.24 for the multi-dimensional case.

Definition I.30. Let U ⊂ L1(G;E), where E =∏pj=1Ei. The set U is said to be uniformly integrable if

the family‖ξ‖1, ξ ∈ U


The following lemma is nothing more that Corollary I.29.(ii),(iv).

Lemma I.31. Let E′ :=∏pj=1Ei and T : (E′, ‖ · ‖1) → (Rdim(E′), ‖ · ‖1) be an isometry that sorts the

elements of E in a row. Then, U ⊂ L1(G;E) is uniformly integrable if and only if(T i(ξ)

)ξ∈U is uniformly

integrable for every i ∈ 1, . . . ,dim(E′), where T i(ξ) denotes the i−element of the vector T (ξ).

Proof. Let U ⊂ L1(G;E) be uniformly integrable, i.e.‖ξ‖1, ξ ∈ U

is uniformly integrable. Then,

|T i(ξ)| ≤ ‖ξ‖1, for every i ∈ 1, . . . ,dim(E′) and by Corollary I.29.(ii) we can conclude.

Conversely, let U ⊂ L1(G;E) be a family such that(T i(ξ)

)ξ∈U is uniformly integrable for every i ∈

1, . . . ,dim(E′). Then, ‖ξ‖1 =∑dim(E′)i=1 |T i(ξ)|. Therefore, we can conclude by Corollary I.29.(iv).

Theorem I.32 (Vitali’s Convergence Theorem). Let (ξk)k∈N ⊂ L1(G;E) be a sequence of E−valuedrandom variables and ξ∞ be an E−valued random variable such that

P([δ‖·‖1(ξk, ξ∞) > ε

])−−→ 0 for every ε > 0.

The following are equivalent:

(i) The convergence ξkL1(G;E)−−−−−−→ ξ∞.

(ii) The sequence (ξk)k∈N is uniformly integrable.

In either case, ξ∞ ∈ L1(G;E). Moreover, the following are equivalent

(i) The convergence ξkL2(G;E)−−−−−−→ ξ∞ holds.

(ii) The sequence(‖ξk‖22

)k∈N is uniformly integrable.

Proof. See Leadbetter, Cambanis, and Pipiras [45, Theorem 11.4.2].

I.2.1. Generalised Inverses. The concept of generalised inverse of an increasing function or pro-cess appears often in the literature either as the right derivative of the Young conjugate of a Youngfunction, see Definition I.40.(i), or as time change, see e.g. [60, Lemma 0.4.8, Proposition 0.4.9]. We areinterested in both cases.


Definition I.33. Let χ : R+ → R+ be an increasing function. The right-continuous generalised inverseof χ is the function χ−1,r : R+ → R+ defined by

χ−1,r(s) :=

inft ∈ R+, χ(t) > s, if t ∈ R+, χ(t) > s 6= ∅,∞, if t ∈ R+, χ(t) > s = ∅

and the left-continuous generalised inverse of χ is the function χ−1,l : R+ → R+ defined by

χ−1,l(s) :=

inft ∈ R+, χ(t) ≥ s, if t ∈ R+, χ(t) ≥ s 6= ∅,∞, if t ∈ R+, χ(t) ≥ s = ∅.

It is well-known, e.g. see Revuz and Yor [60, Lemma 0.4.8, p. 7], that the right-continuous generalisedinverse of a non-decreasing and right-continuous function is also non-decreasing and right-continuous,which justifies the used term. In the following lemma the new result that it is provided by the author is theright inequality of (vii). Using this simple inequality we will be able to establish Theorem II.14. However,we present the other properties, since we will make use of them in the proof or later in Subsection I.2.2.The interested reader can find their proofs in a slightly more general framework in Embrechts and Hofert[32] and the references therein.

Lemma I.34. Let χ : R+ −→ R+ be a cadlag and increasing function with χ0 = 0.

(i) χ−1,l is caglad and increasing, while χ−1,r is cadlag and increasing.(ii) χ−1,l(s) = χ−1,r(s−) and χ−1,l(s+) = χ−1,r(s).

(iii) s ≤ χ(t) if and only if χ−1,l(s) ≤ t and s < χ(t) if and only if χ−1,r(s) < t.(iv) χ(t) < s if and only if t < χ−1,l(s) and χ(t) ≤ s if and only if t ≤ χ−1,r(s).(v) χ

(χ−1,r(s)

)≥ χ

(χ−1,l(s)

)≥ s, for s ∈ R+, and at most one of the inequalities can be strict.

(vi) For s ∈ χ(R+) := s ∈ R+,∃t ∈ R+ such that χ(t) = s, χ(χ−1,l(s)) = s.(vii) For s such that χ−1,l(s) <∞, we have

s ≤ χ(χ−1,l(s)

)≤ s+ ∆χ(χ−1,l(s)),

where ∆χ(χ−1,l(s)) is the jump of the function χ at the point χ−1,l(s).

Proof. We need to prove that χ(χ−1,l(s)) − s ≤ ∆χ(χ−1,l(s)) for any s such that χ−1,l(s) < ∞.By (vi), when s ∈ χ(R+), we have since χ is increasing

χ(χ−1,l(s))− s = 0 ≤ ∆χ(χ−1,l(s)).

Now if s /∈ χ(R+) and s > χ∞ := limt→∞ χ(t), then χ−1,l(s) =∞, so that this case is automaticallyexcluded. Therefore, we now assume that s /∈ χ(R+) and s ≤ χ∞. Since s /∈ χ(R+), there exists somet ∈ R+ such that s ∈ [χ(t−), χ(t)). Then, we immediately have χ−1,l(s) = t. Hence

s+ ∆χ(χ−1,l(s)) = s+ ∆χ(t) ≥ χ(t) = χ(χ−1,l(s)),

since s ≥ χ(t−).

Lemma I.35. Let g be a non-decreasing sub-multiplicative function on R+, that is to say

g(x+ y) ≤ γg(x)g(y),

for some γ > 0 and for every x, y ∈ R+. Let A be a cadlag and non-decreasing function with associatedBorel measure µA. Then it holds that∫

(0,t]

g(As)µA(ds) ≤ γg

(max

s, A−1,ls <∞

∆ALs

)∫(A0,At]

g(s) ds.

Corollary I.36. Let A ∈ V+(G;R) and g as in Lemma I.35 with the additional assumption that A hasuniformly bounded jumps, say by K. Then there exists a universal constant K ′ > 0 such that∫

(0,t]

g(As(ω))µA(ω)(ds) ≤ K ′∫

(A0(ω),At(ω)]

g(s) ds P− a.s..

The constant K ′ equals γg(K), where γ is the sub-multiplicativity constant of g.


I.2.2. Moderate Young Functions. The main aim of this subjection is to present a brief overviewto moderate Young functions; see Definition I.37 and Notation I.41. The importance of this sub-class ofYoung functions is that the Burkholder–Davis–Gundy Inequality, see Theorem I.101, is valid for them.Moreover, we can make use of the Doob’s Maximal Inequality for a function Ψ whose Young conjugateis moderate; see Theorem I.104 and Definition I.40.(i).

Definition I.37. A function Υ : R+ → R+ is called Young function if it is non-decreasing, convex andsatisfies

Υ(0) = 0 and limx→∞

Υ(x)

x= ∞.

Moreover, the non-negative, non-decreasing and right-continuous function υ : R+ → R+ for which

Υ(x) =

∫[0,x]

υ(z) dz

will be called the right derivative of Υ.

Lemma I.38. Let Υ : R+ → R+ be a Young function, then its right derivative exists and is unbounded.

Proof. The existence and the properties of the right derivative of Υ, called υ, is a well-known resultof convex analysis, e.g. see Rockafellar [61, Theorem 23.1, Theorem 24.1, Corollary 24.2.1]. Assume,now, that υ is bounded by a positive constant C. Then

supx∈R+

Υ(x)

x= sup

x∈R+

1

x

∫[0,x]

υ(z) dz ≤ supx∈R+

Cx

x= C < ∞,

which contradicts to the property limx→R+

1xΥ(x) = ∞.

For our convenience, let us collect all the Young functions in a set.

Notation I.39. YF := Υ : R+ → R+,Υ is a Young function.In view of the previous lemma it is immediate that the right-continuous generalised inverse of υ is

real-valued and unbounded. These are properties which justify the following definition.

Definition I.40. (i) The Young conjugate operator : YF −−→ YF is defined by

YF Υ −→ Υ(·) :=∫[0,·]

υ−1,r(z) dz ∈ YF ,

where υ is the right derivative of the Young function Υ and υ−1,r is the right-continuous generalisedinverse of υ; see Definition I.33.

(ii) The conjugate index operator : [1,∞] −−→ [1,∞] is defined by

[1,∞] ϑ −→ ϑ =

⎧⎪⎨⎪⎩∞, if ϑ = 1ϑ

ϑ−1 , if ϑ ∈ (1,∞)

1, if ϑ = ∞

⎫⎪⎬⎪⎭ ∈ [1,∞].

Notation I.41. To a Young function Υ with right derivative υ we associate the constants

cΥ:= inf

x>0

xυ(x)

Υ(x)and sc

Υ:= sup

x>0

xυ(x)

Υ(x).

Remark I.42. Let Υ ∈ YF with right derivative υ. Then for every x ∈ R+

Υ(x) =

∫[0,x]

υ(z) dz ≤ xυ(x), which impliesxυ(x)

Υ(x)≥ 1.

Therefore scΥ , cΥ ≥ 1.

Definition I.43. A Young function Υ is said to be moderate if scΥ < ∞. The set of all moderate Young

functions will be denoted by YFmod.

Lemma I.44. Let Υ ∈ YF . Then

(i) For every x, y ∈ R+ holds xy ≤ Υ(x) + Υ(y). This inequality is called Young’s Inequality.(ii) Υ ∈ YFmod if and only if Υ(λx) ≤ λsc

ΥΥ(x) for every x ∈ R+ and for every λ ≥ 1.(iii) Υ ∈ YFmod if and only if there exists C2 > 0 such that Υ(2x) ≤ C2Υ(x) for every x ∈ R+.


(iv) If scΥ< ∞, then the function R+ x → Υ(x)/xsc

Υ is non-increasing. Inversely, given the constantcΥ, the function R+ x → Υ(x)/xc

Υ is non-decreasing.(v) (c

Υ) = sc

Υ and (scΥ) = c

Υ .

Proof. (i) This is a well-known result, see e.g. Long [48, Theorem 3.1.1 (a)].(ii) See [48, Theorem 3.1.1 (c)-(d)].(iii) In view of (ii) it is only left to prove the sufficient direction, i.e. assuming that there exists C2 > 0

such that Υ(2x) ≤ C2Υ(x) for every x ∈ R+, we need to prove that Υ is moderate. For this see He et al.[35, Definition 10.32, Lemma 10.33.2)].(iv) See [48, Theorem 3.1.1 (e)](v) See [48, Theorem 3.1.1 (f)]. The previous lemma provides us with an easy criterion for proving that the Young conjugate of a

Young function is moderate.

Corollary I.45. Let Υ ∈ YF , then Υ ∈ YFmod if and only if cΥ> 1.

Proof. By the definition of the conjugate indices we have that scΥ < ∞ if and only if c

Υ> 1.

Example I.46. For ϑ ∈ (1,∞) we define R+ xΥϑ−→ 1

ϑxϑ ∈ R+. Let us now fix a ϑ ∈ (1,∞). Then the

derivative of Υϑ is R+ xυϑ−→ xϑ−1 ∈ R+ and we can easily calculate that sc

Υϑ= c

Υϑ= ϑ. Moreover,

υϑis continuous and increasing, therefore its right-continuous generalised inverse is the usual inverse

function. Then, we have directly that υ−1,rϑ

(x) = x1

ϑ−1 , for every x ∈ R+. Consequently,

(Υϑ)(x) =

∫[0,x]

υ−1,rϑ

(z) dz =

∫[0,x]

z1

ϑ−1 dz =ϑ− 1

ϑx

ϑϑ−1 =

1

ϑxϑ

= Υϑ(x), for every x ∈ R+.

Observe, moreover, that Υϑ ∈ YFmod for every ϑ ∈ (1,∞).

It is the next lemma that enables us to always choose a moderate Young function when we applyTheorem I.27.

Lemma I.47 (Meyer). Let U ⊂ L1(G) be a uniformly integrable set. Then there exists Φ ∈ YFmod suchthat the set Φ(|X|), X ∈ U is uniformly integrable.

Proof. See Meyer [52, Lemme, p. 770]. Now we can provide provide the de La Vallee Poussin–Meyer criterion. We urge the reader to compare

with Theorem I.27.

Corollary I.48 (de La Vallee Poussin–Meyer Criterion). A set U ⊂ L1(G) is uniformly integrable if andonly if there exists Φ ∈ YFmod such that supX∈U E[ΦA(|X|)] < ∞.

Proof. The sufficient direction is immediate, since a moderate Young function satisfies the require-ments of Theorem I.27.

For the necessary direction, by Lemma I.47 there exists a moderate sequence A such that the setΦA(|X|), X ∈ U is uniformly integrable. Then, by Theorem I.25.(i) we can conclude.

Let us return for a while to Theorem I.27 and have a closer look to its proof; see Dellacherie andMeyer [26, Theorem II.22, p. 24]. Following the notation of [26], we can see that the function G can berepresented as a Lebesgue integral whose integrand g is a non-decreasing, piecewise-constant, unboundedand positive function, i.e. the function G can be assumed in particular convex. Moreover, the values of gform a non-decreasing and unbounded sequence (gn)n∈N, which is suitably chosen such that the conditionof Theorem I.27 is satisfied. The improvement of Meyer [52, Lemme, p. 1] (now we follow the notation of[52]) consists in constructing a piecewise constant integrand f such that f(2t) ≤ 2f(t). This implies that∫

[0,2x]

f(z) dz =

∫[0,x]

2f(2t) dt ≤ 4

∫[0,x]

f(t) dt

i.e. the Young function with right derivative f is moderate; see Lemma I.44.(iii). The aforementionedproperty of f is equivalent to f2n ≤ 2fn for every n ∈ N, where fn is the value of the piecewise constantf on the interval [n, n+ 1).

We have made the above discussion in order to underline the importance of a sequence (fk)k∈N withthe aforementioned property (among others9) and to justify the following notation.

9The sequence (fn)n∈N∪0 can be chosen such that f0 = 1, fn ∈ N for every n ∈ N ∪ 0 and non-decreasing.


Notation I.49. (i) Let U ⊂ L1(G;E) be a uniformly integrable set and ΦU be a moderate Youngfunction obtained by the de La Vallee Poussin–Meyer criterion. Then, we can associate to the functionΦU , hence also to the uniformly integrable set U , a sequence AU := (αk)k∈N which satisfies

[Mod] For every k ∈ N holds αk ∈ N, αk ≤ αk+1, α2k ≤ 2αk and limk→∞ αk =∞.(ii) A sequence A which satisfies the properties [Mod] will be called moderate sequence and the asso-

ciated moderate Young function will be

ΦA(x) :=

∫ x

0

1[0,1)(t) +∞∑k=1

αk1[k,k+1)(t) dt, for x ∈ R+. (I.2)

The right derivative of ΦA will de denoted by φA .

Remark I.50. It has been already checked that a moderate sequence A associates to a moderate Youngfunction. However, the converse is not true, e.g. the right derivative of quad (recall that it has beendefined on p. 2) is Id.

If we may anticipate into Chapter III, the coming Proposition I.51 is the tool that will allow us toprove Lemma III.39. The latter will be crucial for obtaining sufficient integrability for the weak limits ofthe orthogonal martingales. There we make use of the Burkholder–Davis–Gundy Inequality in the formof Theorem I.101 as well as of Doob’s Inequality in the form of Theorem I.104. In order to apply both ofthe above inequalities for the same Young function Φ, we need Φ to be moderate with moderate Youngconjugate and, additionally, Φ must be (suitably) dominated by a quadratic function. But since it is tooearly for providing more details around that point, let us explain the intuition behind Proposition I.51.

The function quad will play a crucial role in the following. Recall that in its definition we requiredthe presence of the fraction 1

2 so that quad? = quad. This is straightforward since the right derivative of

quad is simply Id and we have immediately that Id−1,r = Id. Let us fix for the next lines two arbitraryYoung functions Φ1,Φ2 such that Φ1(x) ≤ Φ2(x) for large x ∈ R+. Then it is well-known, e.g. seeKrasnosel’skii and Rutickii [43, Theorem 2.1, p. 14] or Rao and Ren [58, Proposition I.2], that for theirYoung conjugates Φ?1,Φ

?2 holds Φ?2(x) ≤ Φ?1(x) for large x ∈ R+. Assume, moreover, that Φ1 is moderate,

then it is straightforward to prove that Φ1 quad is also moderate and due to the growth of Φ1 we canconclude that quad(x) ≤ Φ1 quad(x) for large x ∈ R+, where Φ1 quad(x) := Φ1(quad(x)), for x ∈ R+.Therefore, in view of the previous comments, we have that (Φ1 quad)?(x) ≤ quad(x). Proposition I.51verifies that, if we choose Φ1 to be a moderate Young function associated to a moderate sequence A, thenthe Young function (Φ1 quad(x))? has nice properties, namely it is moderate and, moreover, accepts aYoung function Υ such that Υ (Φ1 quad)? = quad.

Proposition I.51. Let A := (αk)k∈N be a moderate sequence and define ΦA,quad := ΦA quad. Let,moreover, Ψ := (ΦA,quad)? and ψ be the right derivative of Ψ. Then

(i) ΦA,quad,Ψ ∈ YFmod.(ii) ψ is continuous and can be written as a concatenation of linear and constant functions defined

on intervals. The slopes of the linear parts consist a non–increasing and vanishing sequence of positivenumbers.

(iii) It holds Ψ(x) ≤ quad(x), where the equality holds on a compact neighbourhood of 0, and

limx↑∞

(quad(x)−Ψ(x)) =∞.

(iv) There exists Υ ∈ YF such that Υ Ψ = quad.

Proof. (i) We will prove initially that ΦA,quad is a Young function. It is sufficient to prove thatit can be written as a Lebesgue integral whose integrand is a cadlag, increasing and unbounded function;see Definition I.37 and Lemma I.38.

For every x ∈ R+ holds by definition

ΦA,quad(x) = ΦA(1

2x2) =

∫[0, 12x

2]

φA(z) dz =

∫[0,x]

tφA(1

2t2) dt. (I.3)

We define φA,quad: R+ → R+ by

φA,quad(t) := tφA(

1

2t2) = t1[0,

√2)(t) + t

∞∑k=1

αk1[√

2k,√

2k+2)(t), (I.4)

i.e. φA,quadis cadlag and piecewise-linear. Observe, moreover, that


ΔφA,quad(√2) = (α1 − 1)

√2 ≥ 0.

ΔφA,quad(√2k + 2) = (αk+1 − αk)

√2k + 2 ≥ 0, for every k ∈ N.

φA,quadhas increasing slopes; the value of the slope of the linear part defined on the interval

[√2k,

√2k + 2) is determined by the value of the respective element αk ≥ 1, for every k ∈ N.

lims→∞ φA,quad(s) = ∞.

Therefore, ΦA,quad is a Young function and its conjugate Ψ is also a Young function.

We will prove now that both ΦA,quad and Ψ are moderate. We have directly that cquad = scquad = 2.

Moreover, by the property α2k ≤ 2αk we have that ΦA is moderate, hence scΦA < ∞. Now we obtain

cΦA,quad= inf

x>0

xφA,quad(x)

ΦA,quad(x)= inf

x>0

x2φA(12x

2)

ΦA( 12x2)

= 2 infx>0

12x

2φA(12x

2)

ΦA( 12x2)

= 2 infu>0

uφA(u)

ΦA(u)= 2cΦA ≥ 2,

because for every Young function Υ ∈ YF holds cΥ

≥ 1; see Remark I.42. For scΦA,quadwe have

scΦA,quad= sup

x>0

xφA,quad(x)

ΦA,quad(x)= 2 sup

u>0

uφA(u)

ΦA(u)= 2scΦA < ∞. (I.5)

Hence, ΦA,quad is a moderate Young function. Moreover, since cΦA,quad> 1, we have from Corollary I.45

that Ψ is moderate.

(ii) For the rest of the proof, i.e. for parts (ii)-(iv), we will simplify the notation and we will write φfor the function φA,quad

.

Firstly, let us observe that ψ is real-valued, resp. unbounded, since φ is unbounded, resp. real-valued.10 In order to determine the value ψ(s) for s ∈ (0,∞) let us define two sequences of subsets of R+

(Ck)k∈N∪0, (Jk)k∈N∪0 by

Ck :=[φ(

√2k), lim

t↑√2k+2φ(t))and Jk+1 :=

[lim

t↑√2k+2φ(t), φ(

√2k + 2)

), for k ∈ N ∪ 0. (I.6)

Observe that Ck = φ

([√2k,

√2k + 2)

) = ∅ for every k ∈ N ∪ 0, because φ is continuous and increasing on

[√2k,

√2k + 2).

Jk = ∅ if and only if φ is continuous at√2k + 2, which is further equivalent to αk = αk+1.

For our convenience, let us define two sequences (sk)k∈N∪0, (sk+1− )k∈N∪0 of positive numbers as follows:

s0 := 0, sk := φ(√2k) = αk

√2k, for k ∈ N and

sk+1− := lim

x↑√2k+2φ(x) = αk

√2k + 2, for k ∈ N ∪ 0. (I.7)

The introduction of the last notation permits us to write Ck = [sk, sk+1− ) and Jk = [sk+1

− , sk+1), fork ∈ N ∪ 0. Now we are ready to determine the values of ψ on (0,∞). The reader should keep in mindthat the function φ is increasing and right-continuous.

• Let s ∈ C0 =[φ(0), φ(

√2−))= φ([0,

√2)), then

ψ(s) = inft ∈ R+, φ(t) > s s∈φ([0,√2))

= inft ∈ [0,

√2), φ(t) > s

(I.4)= inft ∈ R+, Id(t)1[0,

√2)(t) > s = s,

where the second equality is valid because φ is continuous on [0,√2) with φ

([0,

√2))= [0, s1−). To sum

up, ψ1[s0,s1−) = Id1[s0,s1−).

• Let s ∈ J1 =[φ(

√2−), φ(

√2))= [s1−, s

1). If J1 = ∅, which amounts to α1 = 1, there is nothing

to prove. On the other hand, if J1 = ∅, then φ(√2−) ≤ s < φ(

√2) and consequently

ψ(s) = inft ∈ R+, φ(t) > s =√2 for every s ∈ J1.

To sum up, ψ1[s1−,s1) =√21[s1−,s1).

For the general case let us fix a k ∈ N. We will distinguish between the cases s ∈ Ck and s ∈ Jk.For the latter we can argue completely analogous to the case s ∈ J1, but for the sake of completeness wewill provide its proof.

10The reader who is not familiar with generalised inverses may find helpful Embrechts and Hofert [32], especially thecomments after [32, Remark 2.2].


• Let s ∈ Ck =[φ(√

2k), φ(√

2k + 2−))

= φ([√

2k,√

2k + 2)). Since the image of [

√2k,√

2k + 2)

through R+ 3 tχ7−→ αkt ∈ R+ is Ck, then ψ has to coincide with R+ 3 s

χ−1,r

7−−−−−→ 1αks ∈ R+ on

Ck = [sk, sk+1− ). To sum up, ψ(s)1[sk,sk+1

− ) = 1αk

Id1[sk,sk+1− ).

• Let s ∈ Jk+1 =[φ(√

2k + 2−), φ(√

2k + 2))

= [sk+1− , sk+1). If Jk+1 = ∅, which amounts to αk+1 =

αk, there is nothing to prove. On the other hand, if Jk+1 6= ∅, then φ(√

2k + 2−) ≤ s < φ(√

2k + 2) andconsequently

ψ(s) = inft ∈ R+, φ(t) > s =√

2k + 2 for every s ∈ Jk+1.

To sum up, ψ1[sk+1− ,sk+1) =

√2k + 21[sk+1

− ,sk+1).

In total we have that the right derivative of Ψ can be written as a concatenation of linear and constantfunctions defined on intervals

ψ(s) = Id1[s0,s1−) +∞∑k=1

1

αkId1[sk,sk+1

− ) +∞∑k=0

√2k + 21[sk+1

− ,sk+1) (I.8)

where the interval end-points are defined in (I.7). Recall now that αk ≥ 1, for every k ∈ N, therefore wehave that the slopes of ψ are smaller than 1. Moreover, since αk ≤ αk+1 and limk→∞ αk = ∞, we havethat 1

αk+1 ≤ 1αk

and limk→∞1αk

= 0. Finally, as it can be easily checked, ψ is continuous. This causes no

surprise, since φ is strictly increasing; see Embrechts and Hofert [32, Proposition 2.3.(7)].11

(iii) Let us consider the function ζ : R+ → R defined by ζ := Id−ψ, i.e. ζ is also continuous. Moreover,

ζ is differentiable on a superset of R+ \(sk)k∈N∪0∪ (sk+1− )k∈N∪0, which is clearly an open and dense

subset of R+ since there is no accumulation point in the sequence (sk)k∈N∪0∪(sk+1− )k∈N∪0. Obviously

ζ ′(x) =

1− 1

αk, for x ∈ (sk, sk+1

− ) for k ∈ N ∪ 0,1, for x ∈ (sk+1

− , sk+1) for k ∈ N ∪ 0, whenever (sk+1− , sk+1) 6= ∅.

Define M := mink ∈ N, αk > 1, which is a well-defined positive integer, since αk −−→ ∞. Recall nowthat αk ≥ 1 for k ∈ N and we can conclude that ζ ′(x) > 0 almost everywhere on [sM ,∞).

We prove now that quad and ψ coincide only on a compact neighbourhood of 0. By the definition ofM we have that αk = 1 for k ∈ 1, . . . ,M − 1 and αM > 1, therefore Id1[0,sM ] = ψ1[0,sM ] and x > ψ(x)

for every x ∈ (sM ,∞).

Finally, it is left to prove that limx↑∞(Id(x)−ψ(x)) =∞. Recall that (sk+1− , sk+1) 6= ∅ whenever k is

such that αk < αk+1, i.e. there are infinitely many non-trivial intervals (sk+1− , sk+1). But these intervals

correspond to the intervals that ψ is constant. Since Id is increasing we can conclude that ζ is unboundedand therefore also that the required limit is valid.

(iv) For the following recall (I.7), (I.8) and the definition of M. Let us start with the introduction of the

auxiliary function η : ∪∞k=M

[Ψ(sk+1

− ),Ψ(sk+1))−−→ (0, 1] defined by η(z) := Ψ(sk+1)−z

Ψ(sk+1)−Ψ(sk+1− )

.12 Recall

now that Ψ is continuous and increasing, which allows us to define the function υ : R+ → R+ by

υ(z) = 1[0,Ψ(sM ))(z) +∞∑

k=M

αk1[Ψ(sk),Ψ(sk+1− ))(z)

+∞∑

k=M

(η(z)αk + (1− η(z))αk+1

)1[Ψ(sk+1

− ),Ψ(sk+1))(z).

(I.9)

We can directly check that υ is indeed well-defined, non–negative, non–decreasing and unbounded. There-fore, the function υ is the right derivative of a Young function, call Υ.

We intend to prove that Υ Ψ = quad, which is equivalent to proving that the right-derivative ofΥ Ψ equals Id. The following simple calculations allow us to evaluate the right derivative of Υ Ψ interms of υ, ψ and Ψ. For x ∈ R+ → R+ holds

Υ Ψ(x) =

∫[0,Ψ(x)]

υ(t) dt =

∫[0,x]

ψ(z)υ(Ψ(z)

)dz.

11In [32] the presented results regard the left-continuous generalised inverse of a function. However, by Lemma I.34.(ii)

we can conclude also for the right-continuous generalised inverse.12In fact, η(z) is the unique number in [0, 1) for which z can be written as convex combination of Ψ(sk+1

− ) and Ψ(sk+1).


Now we can compare the right derivative of Υ Ψ, which is the function ψ(υ Ψ) : R+ → R+, with theidentity function Id. To this end, we will consider the behaviour of ψ(υ Ψ) on the intervals [0, sM ),

[sk, sk+1− ), for k ≥M and [sk+1

− , sk+1) for k ≥M, which consist a partition of R+. Before we proceed, letus evaluate the function υ at Ψ(s) for s ∈ R+ :

υ(Ψ(s)

)= 1[0,Ψ(sM ))(Ψ(s)) +

∞∑k=M

αk1[Ψ(sk),Ψ(sk+1− ))(Ψ(s))

+∞∑

k=M

η(Ψ(s)

)αk +

[1− η

(Ψ(s)

)]αk+1

1[Ψ(sk+1

− ),Ψ(sk+1))(Ψ(s))

= 1[0,sM )(s) +∞∑

k=M

αk1[sk,sk+1− )(s)

+∞∑

k=M

η(Ψ(s)

)αk +

[1− η

(Ψ(s)

)]αk+1

1[sk+1− ,sk+1)(s) (I.10)

because Ψ is continuous and increasing.

• Let s ∈ [0, sM ). At the end of the proof of (iii) we obtained that Id1[0,sM ) = ψ1[0,sM ). Therefore,we can conclude that

ψ(s)(υ Ψ)(s) = sυ(Ψ(s)

)= s.

• Let s ∈ [sk, sk+1− ), for some k ≥M. Then,

ψ(s)υ(Ψ(s))(I.8)=

(I.10)

1

αkαks = s.

• Let s ∈ [sk+1− , sk+1) for some k ≥M. Then, for the chosen s there exists (unique) µs ∈ [0, 1) such

that

s = µssk+1− + (1− µs)sk+1. (I.11)

But, Ψ is linear on [sk+1− , sk+1), i.e.

Ψ(s) = µsΨ(sk+1− ) + (1− µs)Ψ(sk+1) (I.12)

and Ψ(s) ∈[Ψ(sk+1

− ),Ψ(sk+1)). Therefore, by definition of η

η(Ψ(s)) =Ψ(sk+1)−Ψ(s)

Ψ(sk+1)−Ψ(sk+1− )

(I.12)=

Ψ(sk+1)− µsΨ(sk+1− )− (1− µs)Ψ(sk+1)

Ψ(sk+1)−Ψ(sk+1− )

=Ψ(sk+1)− µsΨ(sk+1

− )−Ψ(sk+1) + µsΨ(sk+1)

Ψ(sk+1)−Ψ(sk+1− )

= µs (I.13)

and finally, in view of

ψ(s)(υ Ψ)(s) =√

2k + 2 υ(Ψ(s))(I.10)

=√

2k + 2η(Ψ(s)

)αk +

[1− η

(Ψ(s)

)]αk+1

(I.13)

=√

2k + 2(µsαk + (1− µs)αk+1)

(I.7)= µss

k+1− + (1− µs)sk+1 (I.11)

= s,

we can conclude.

In the rest of this section we will state some classical results, which shed more light on the niceproperties of a moderate Young function. Before we proceed we provide the necessary notation.

Notation I.52. Let Υ ∈ YF . In the following, ξ denotes a G−measurable and real-valued variable.

• LΥ(G) := ξ, E[Υ(|ξ|)

]<∞

• LΥ(G) := ξ, exists λ > 0 such that E [Υ(|ξ|/λ)] <∞.• ‖ξ‖Υ := infλ > 0,E [Υ(|ξ|/λ)] ≤ 1, where inf ∅ :=∞.

Proposition I.53. Let Υ ∈ YF .

(i) A sufficient condition for the set LΥ to be vector subspace of L1(G;R) is Υ to be moderate.(ii) The set LΥ(G) is a vector subspace of L1(G;R).


(iii) The functional ‖ · ‖Υ : LΥ −−→ R+ is a norm.13

(iv) The normed space (LΥ(G), ‖ · ‖Υ) is Banach.

(v) If Υ ∈ YFmod, then the vector spaces LΥ and LΥ are the same.

Proof. (i) See Rao and Ren [58, Subsection III.3.1 Theorem 2.(ii), p. 46] and [58, SubsectionII.2.3, Definition 1] for the notation.

(ii) See [58, Subsection III.3.1 Proposition 6, p. 49].(iii) See [58, Subsection III.3.1 Theorem 3, p. 54].(iv) See [58, Subsection III.3.3 Theorem 10, p. 67].

(v) Let ξ ∈ LΥ(G), i.e. E[Υ(|ξ|)] < ∞, which implies that ‖ξ‖Υ ≤ 1 ∨ E

[Υ(|ξ|)], where we have used

the convexity of Υ and the fact that Υ(0) = 0. For the converse we need the Young function Υ to bemoderate. Let ξ ∈ LΥ(G), then

E[Υ(|ξ|)] = E

[Υ(‖ξ‖Υ |ξ|

‖ξ‖Υ)]

≤E[Υ(|ξ|/‖ξ‖Υ)] if ‖ξ‖Υ ≤ 1,

sc‖ξ‖ΥΥ

E[Υ(|ξ|/‖ξ‖Υ)] otherwise,

(I.14)

where in the first case we used the convexity of the Young function Υ, while in the second Lemma I.44.(ii).

Therefore, since by definition of ‖ξ‖Υ we have E

[Υ(

|ξ|‖ξ‖Υ

)]≤ 1, in either case ξ ∈ LΥ(G).

In view of the previous result the following definition is allowed.

Definition I.54. Let Υ ∈ YF . The set LΥ(G) will be called Orlicz space.

We close this section with the following very convenient result.

Theorem I.55. Let S ∈ Ssp(G;R) with G−canonical decomposition S = X +A with A non-decreasing,i.e. S is a G−submartingale, and Φ ∈ YFmod. Then

E[Φ(A∞)] ≤ (2cΦ)cΦE[Φ(sup

s>0|Ss|)].

Proof. See Lenglart, Lepingle, and Pratelli [47, Theoreme 3.2.1)].

I.3. Uniformly integrable martingales

In this section the state space is Rp×q, except otherwise stated, and it is denoted by E. We are goingto present results regarding the space of uniformly integrable G−martingales14 as well as results for itssubspace containing all the square integrable G−martingales. Recall that by Notation I.11 the former isdenoted by M(G;E), while the latter by H2(G;E).

We start with the lemma that verifies that indeed H2(G;E) ⊂ M(G;E).

Lemma I.56. If X ∈ H2(G;E), then X ∈ M(G;E).

Proof. We apply Theorem I.27 for the function R+ xΦ−→ x2 ∈ R+. Obviously, Φ ∈ Y. Moreover,

supt∈R+

E[Φ(‖Xt‖1)

] ≤ supt∈R+

E[‖Xt‖21

] ≤ pq supt∈R+

E[‖Xt‖22

]< ∞.

Theorem I.57. Let X be an E−valued stochastic process and X∞(ω) := limt↑∞ Xt(ω), for every ω ∈ Ω.

(i) If X ∈ M(G;E), then the random variable X∞ is P−almost surely well-defined and lies inL1(G∞;E). Moreover,

XtL1(G∞;E)−−−−−−−→ X∞ and Xt = E[X∞|Gt] P− a.s. for every t ∈ R+,

i.e. the G−martingale X is (right) closed by its terminal value X∞.(ii) If X ∈ M(G;E), then X ∈ H2(G;E) if and only if ‖X∞‖L2(G∞;E) < ∞. In this case,

supt∈R+

‖Xt‖L2(G;E) ≤ ‖X∞‖2L2(G;E) ≤ pq sup

t∈R+

‖Xt‖L2(G;E).

13As usual we identify the P− a.s. equal random variables.14Recall that the G−stochastic basis is already given in Section I.1.

I.3. UNIFORMLY INTEGRABLE MARTINGALES 17

Proof. (i) For every pair (i, j), for 1 ≤ i ≤ p, 1 ≤ j ≤ q, we apply Dellacherie and Meyer [27,Theorem VI.6]. More precisely, for every pair (i, j), for 1 ≤ i ≤ p, 1 ≤ j ≤ q, there exists Ωij ⊂ Ω withP(Ωij) = 1 such that Xij

∞(ω) is well-defined for every ω ∈ Ωij ,

Xijt

L1(G∞;R)−−−−−−−→ Xij∞ and Xij

t = E[Xij∞|Gt] P− a.s. for every t ∈ R+.

By the above convergence we obtain

‖Xt −X∞‖L1(G∞;E) =

p∑i=1

q∑j=1

E[∣∣Xij

t −Xij∞∣∣] −−−→

t↑∞0.

Define now ΩE :=⋂Ωij , 1 ≤ i ≤ p and 1 ≤ j ≤ q. Then P(ΩE) = 1 and Xij

∞(ω) is well-defined forevery 1 ≤ i ≤ p, 1 ≤ j ≤ q and for every ω ∈ ΩE . Therefore, also the property Xt = E[X∞|Gt] holdsP−almost surely .

(ii) Let X ∈ H2(G;E), i.e. supt∈R+E[‖Xt‖22

]< ∞. We prove initially that Xij

∞ ∈ L2(G∞;R) for1 ≤ i ≤ p, 1 ≤ j ≤ q. The obvious domination

supt∈R+

E[|Xij

t |2]≤ supt∈R+

E[‖Xt‖22

]<∞

enables us to apply He et al. [35, Theorem 6.8.1] element-wise and, consequently, to obtain that

E[ |Xij∞|2] = sup

t∈R+

E[ |Xijt |2] for 1 ≤ i ≤ p, 1 ≤ j ≤ q. (I.15)

But this is equivalent to X∞ ∈ L2(G;E), more precisely X∞ ∈ L2(G∞;E).Conversely, let X∞ ∈ L2(G∞;E), i.e. E[ |Xij

∞|2] < ∞, for 1 ≤ i ≤ p, 1 ≤ j ≤ q. By He et al. [35,Theorem 6.8.1] applied element-wise we obtain that Equality (I.15) holds. Therefore,

supt∈R+

‖Xt‖2L2(G;E) = supt∈R+

p∑i=1

q∑j=1

E[|Xij

t |2] (I.15)

≤ supt∈R+

p∑i=1

q∑j=1

E[|Xij∞|2]

= ‖X∞‖2L2(G;E). (I.16)

Moreover,

E[‖X∞‖22

]=

p∑i=1

q∑j=1

E[|Xij∞|2] (I.15)

=

p∑i=1

q∑j=1

supt∈R+

E[|Xij

t |2]

= sup

p∑i=1

q∑j=1

E[|Xij

tij|2], tij ∈ R+ for 1 ≤ i ≤ p, 1 ≤ j ≤ q

≤p∑i=1

q∑j=1

(supt∈R+

E[|Xij

t |2])≤

p∑i=1

q∑j=1

(supt∈R+

E[‖Xt‖22

])= pq sup

t∈R+

E[‖Xt‖22

].

The above inequality and (I.16) allow us to conclude.

Remark I.58. In view of the previous theorem, for an X ∈M(G;E) we can write (Xt)t∈R+or (Xt)t∈R+

interchangeably. Observe that, due to the “left-continuity” of X at the symbol ∞ the random variablesupt∈R+

|Xt| is P− a.s. well-defined and P− a.s. equal to supt∈R+|Xt|.

Definition I.59. An E−valued and G ⊗ B(R+)−measurable process M is of class (D) for the filtrationG if the family ‖Mτ‖11[τ<∞], τ is G−stopping time is uniformly integrable.

Theorem I.60 (Doob–Meyer Decomposition). If X is a real-valued, right-continuous G−submartingaleof class (D), then there exists a unique, up to indistinguishability, A ∈ V+

pred(G;R) with A0 = 0 such that

X −A ∈M(G;R).

Corollary I.61. Let X ∈ H2(G;R), then there exists a unique, up to indistinguishability, 〈X〉 ∈V+

pred(G;R) with 〈X〉0 = 0 such that X2 − 〈X〉 ∈ M(G;R).

Proof. It is immediate by Jensen Inequality that the process X2 is a G−submartingale. Then weconclude by Theorem I.60.


Theorem I.62. Let X,Y ∈ H2(G;R). Then, there exists a unique up to indistinguishability process〈X,Y 〉G ∈ Apred(G;R) such that XY − 〈X,Y 〉G ∈M(G;R). Moreover,

〈X,Y 〉G =1

4

(〈X + Y,X + Y 〉G − 〈X − Y,X − Y 〉G

).

Furthermore, 〈X,X〉G is non-decreasing and it admits a continuous version if and only if X is G−quasi-left-continuous.

Proof. See Jacod and Shiryaev [41, Theorem I.4.2].

The process 〈X,Y 〉G is called the predictable quadratic variation or the angle bracket of the pair(X,Y ). When no confusion can arise we will omit the symbol of the filtration from the notation. Moreover,〈X,X〉 will be also denoted as 〈X〉.

Notation I.63. WhenX ∈ H2(G;Rp), Y ∈ H2(G;Rq) we will denote by 〈X,Y 〉 the element ofApred(G;E)for which 〈X,Y 〉ij = 〈Xi, Y j〉.

Theorem I.64. Let A ∈ A(G;R). There exists a process, which will be called the compensator of Aand denoted by A(p,G), which is unique up to an evanescent set, and which is characterised by being anelement of Apred such that A−A(p,G) ∈M(G;R).

Proof. See Jacod and Shiryaev [41, Theorem I.3.18].

Corollary I.65. Let X ∈M(G;R) ∩ Vpred(G;R). Then X is indistinguishable from the zero process.

Proof. See Jacod and Shiryaev [41, Corollary I.3.16].

I.4. Stochastic integration

This section is devoted to introducing the notation and the main results regarding integration withrespect to a finite variation process, a square-integrable martingale and an integer-valued random measure.The reader may recall that we work with the G−stochastic basis, where G is an arbitrary filtration, untilthe end of the chapter.

I.4.1. Stochastic integration with respect to an increasing process. This section is devotedto Lebesgue–Stieltjes integration either with respect to a finite variation function or a process of finitevariation. In the latter case notation has been provided on page v. Nevertheless, we will introduceadditional one which will be used interchangeably. Let a non–negative measure % defined on (R+,B(R+))and G be any finite dimensional space. We will denote the Lebesgue–Stieltjes integral of any measurablemap f : (R+,B(R+)) −→ (G,B(G)) by∫

(u,t]

f(s)%(ds) and

∫(u,∞)

f(s)%(ds), for any u, t ∈ R+.

The above integrals as well as every Lebesgue–Stieltjes integral are to be understood in a component–wise sense. In case % is a finite measure with associated distribution function F %(·) := %([0, ·]), we willindifferently denote the above integrals by∫

(u,t]

f(s)dF %s and

∫(u,∞)

f(s)dF %s , for any u, t ∈ R+.

When there is no confusion as to which measure the distribution function F % is associated to, we willomit the upper index and we will write F . Conversely, if the distribution function is given, say F , thenwe will denote the associated Borel measure by µF .

When % is a signed measure with Jordan–Hahn decomposition % = %+ − %−, then we define∫(u,t]

f(s)%(ds) :=

∫(u,t]

f(s)%+(ds)−∫

(u,t]

f(s)%−(ds) for any u, t ∈ R+.

and analogously for∫(u,∞)

f(s)%(ds) :=

∫(u,∞)

f(s)%+(ds)−∫

(u,∞)

f(s)%−(ds) for any u ∈ R+.

I.4. STOCHASTIC INTEGRATION 19

If V % is the associated to the signed measure % finite variation function we will denote interchangeablythe above integrals by ∫

(u,t]

f(s) dV %s and

∫(u,∞)

f(s)dV %s for any u, t ∈ R+.

More generally, for any measure % on (R+ × E,B(R+) ⊗ B(E)) and for any measurable map g :(R+ × E,B(R+)⊗ B(E)) −→ (G,B(G)) we will denote the Lebesgue–Stieltjes integral by∫

(u,t]×Ag(s, x)%(ds,dx) and

∫(u,∞)×A

g(s, x)%(ds,dx), for any t, u ∈ R+, A ∈ B(E).

Conversely, if A ∈ V+(G) is given, then we will denote the associated to A(ω) Borel measure by µA(ω), forevery ω ∈ Ω. Then, µA is P−almost surely well-defined. Finally, the integration with respect to a finitevariation process will be understood pathwise. More precisely, if A ∈ V(G;R) and H is an E−valuedrandom function such that H(ω, ·) is Borel measurable, then we define the random integral, denoted by∫

(0,·]Hs dAs, as follows:∫(0,·]

Hs dAs(ω) :=

∫(0,·]Hs(ω) dAs(ω) if

∫(0,·] ‖Hs(ω)‖1 d

[Var(A)s(ω)

]<∞,

∞ otherwise.

Clearly, if H is assumed G−optional, then∫

(0,·]Hs dAs is a process.

I.4.2. The Ito stochastic integral. In this section we present the main results concerning Itostochastic integration with respect to an `−dimensional square-integrable martingale. We follow Jacodand Shiryaev [41, Section III.6.a]. We assume that the real-valued case is familiar to the reader. Ifnot, they may consult Jacod and Shiryaev [41, Section I.4d]. For the following, the arbitrary processX ∈ H2(G;R`) is fixed. In Section I.3 we have seen that the R`×`−valued process 〈X〉 is well-definedand such that 〈X〉 ∈ A(G;R`×`). Consider, now, a factorization of 〈X〉, that is to say, write

〈X〉· =

∫(0,·]

d〈X〉sdFs

dFs, (I.17)

where F is an increasing G−predictable and cadlag process and(

d〈X〉·dF·

)ij:=

d〈X〉ij·dF·

for 1 ≤ i, j ≤ `. It

can be proven by standard arguments that(d〈X〉·

dFs·)t∈R+

is a G−predictable process with d〈X〉tdFt

being a

symmetric, non-negative definite element of R`×`, for every t ∈ R+.

Notation I.66. We will denote by

• H2(G, X;Rp×`) :=Z : (Ω× R+,PG) −−→ (Rp×`,B(Rp×`)), ‖Z‖H2(G,X;R`) <∞

,

where ‖Z‖2H2(G,X;Rp×`) := E[ ∫

(0,∞)

Tr[Zs

d〈X〉sdFs

Z>s

]dFs

].

The following theorem verifies that the above space provides a suitable class of integrands for the Itointegral with respect to an `−dimensional X.

Theorem I.67. For every Z ∈ H2(G, X;Rp×`) the Ito stochastic integral of Z with respect to X underthe filtration G, denoted indifferently by Z ·X or by

∫ ·0ZsdXs,

15 is a well-defined Rp×1−valued process.Moreover,

(i) If Z ∈ H2(G, X;Rp×`) and bounded then

(Z ·X)i =∑j=1

Zij ·Xj for every i=1,. . . ,p.

(ii) For every Z ∈ H2(G, X;Rp×`), there exists a sequence (Zk)k∈N ⊂ H2(G, X;Rp×`), where Zk isbounded for every k ∈ N, such that∥∥Zk ·X∞ − Z ·X∞∥∥L2(G;Rp)

−−−−→k→∞

0.

(iii) (Z ·X)> ∈ H2(G;Rp), for every Z ∈ H2(G, X;Rp×`).15We have abstained from using the transpose of X in the notation.


(iv) If Z1, Z2 ∈ H2(G, X;R`) then

〈Z1 ·X,Z2 ·X〉· =

∫(0,·]

Z1s

d〈X〉sdFs

(Z2s )> dFs.

(v) Let Y ∈ H2(G;R) and define the R`−valued G−predictable process rY := (rY 1, . . . , rY `) by

〈Y,Xi〉· =

∫(0,·]

rY is dFs, for every i = 1, . . . , `.

Then, for Z ∈ H2(G, X;R`), Z ·X is the unique, up to indistinguishability, square-integrable martingalewith Z ·X0 = 0 such that

〈Z ·X,Y 〉· =

∫(0,·]

Zs(rYs )>dFs, for every Y ∈ H2(G;R).

(vi) For Z ∈ H2(G;Rp×`), the jump process of (Z · X)i is indistinguishable from∑`n=1 Z

in∆Xn. In

other words, the i−element of the purely discontinuous part of Z ·X can be described by∑`n=1 Z

in ·Xd,n.This allows us to denote (Z ·X)d by Z ·Xd.(vii) If X ∈ H2,c(G;R`), then for every Z ∈ H2(G, X;R`) holds Z ·X ∈ H2,c(G;R).

Proof. We use Jacod and Shiryaev [41, Theorem III.6.4] to verify that (Z ·X)i is well-defined, fori = 1, . . . , p. For (ii) we use again element-wise [41, Theorem III.6.4 a)]. For (vi) see Jacod and Shiryaev[41, Proposition III6.9]. The other parts are the same as in [41, Theorem III.6.4 a)], but in a differentorder.

Notation I.68. • To the class H2(G, X;Rp×`) we associate the space L2(G, X;Rp) where

L2(G, X;Rp) :=W ·X,W ∈ H2(G, X;Rp×`)

. (I.18)

In other words, L2(G, X;Rp) is the space of the Rp−valued Ito stochastic integrals with respect to thepair (G, X).• When F is a filtration such that X ∈ H2(F;R`), in other words when X is both G− and F− adapted,

for a Z ∈ H2(F, X;Rp×`), we will denote the Ito stochastic integral of Z with respect to X under thefiltration G, resp. under the filtration F, by Z ·XG, resp. Z ·XF.

We close this subsection with the following well-known result.

Theorem I.69 (Galtchouk–Kunita–Watanabe Decomposition). Let X ∈ H2(G;R`) and Y ∈ H2(G;Rp).Then, there exists Z ∈ H2(G, X;Rp×`) such that

Y j = (Z ·X)j +N j for every j = 1, . . . , p,

where 〈N j , Xi〉 = 0 for i = 1, . . . , ` and j = 1, . . . , p. The stochastic integral Z ·X does not depend uponthe chosen version of Z.

Proof. See Jacod [40, Chapitre IV, Section 2].

I.4.3. Stochastic integral with respect to a random measure. In this section we describe theconstruction of the stochastic integral with respect to a compensated integer-valued random measure andwe present the results we are going to use in the next chapters. We will follow closely Jacod and Shiryaev[41, Section II.1]. However, Cohen and Elliott [20, Chapter 13] and He et al. [35, Chapter 11] will be alsouseful. The state space will be assumed to be E, except otherwise explicitly stated.

Definition I.70. A random measure µ on R+ × R` is a family µ =(µ(ω; dt,dx)

)ω∈Ω

of measures on(R+ × R`,B(R+)⊗ B(R`)

)satisfying µ(ω; 0 × R`) = 0 identically.

We proceed to introduce some notation and definitions.

Notation I.71. Write

• Ω := Ω× R+ × R`.• (Ω, PG) := (Ω,PG⊗B(R`)

). A measurable function U : (Ω, P) −−→ (E,B(E)) will be said G−predictable

function• (Ω, O) := (Ω,O ⊗ B(R`)

)A measurable function U : (Ω, O) −−→ (E,B(E)) will be said G−optional

function


Definition I.72. Given a random measure µ and a G−optional function W we define the stochasticintegral of W with respect to the random measure µ to be the E−valued process

U ∗ µ·(ω) :=

∫

(0,·]W (ω, t, x)µ(ω; dt,dx), if

∫(0,·]

∥∥W (ω, t, x)∥∥

1µ(ω; dt,dx) <∞,

∞, otherwise.

Observe that we have defined the integral ω−wise. This is indeed eligible, because W is assumed to beG−optional function. Therefore, W (ω, ·) is B(R+)⊗ B(R`)−measurable for each ω. Observe, moreover,that for a real-valued and G−optional function W which is positive, then W ∗ µ ∈ V+(G;R).

Remark I.73. This point is also a good chance to adopt a convenient abuse of notation. Let a function

h :(R`,B(R`)

)−−→

(E,B(E)

). We will treat h also as the G−predictable function Ω 3 (ω, t, x)

h7−→h(x) ∈ E. Analogously, if A is an E−valued G−predictable process, resp. G−optional process, we will

treat it also as Ω 3 (ω, t, x) 7−→ At(ω) ∈ E, which is a G−predictable function, resp. a G−optionalfunction.

Definition I.74. A random measure is called G−optional , resp. G−predictable, if the process W ∗ µ isG−optional process, resp. G−predictable process, for every G−optional, resp. G−predictable, functionW .

We introduce now the set from which we will choose the random measures we are going to work with.

Definition I.75. A G−optional random measure will be said G−predictably σ−integrable or interchange-

ably PG − σ−finite if there exists a positive G−predictable function V such that V ∗ µ ∈ A+(G;R). The

set of G−optional G−predictably σ−integrable random measures will be denoted by Aσ(G).

Observe that an equivalent condition for a random measure to lie in Aσ(G) is the existence of a

partition of PG, say (Ak)k∈N, such that 1Ak ∗ µ ∈ A+(G;R) for every k ∈ N.

Definition I.76. A random measure µ will be called integer-valued G−random measure if it satisfies

(i) µ(ω; t × R`) ≤ 1 identically.(ii) For each A ∈ B(R+)⊗ B(R`), the random variable µ(·;A) takes values in N ∪ 0.

(iii) µ ∈ Aσ(G).

In the remaining chapters we will be interested in random measures associated to the jumps ofcadlag processes. The next proposition verifies that to every G−adapted, cadlag and R`−valued processassociates an integer-valued G−random measure.

Proposition I.77. Let X be a G−adapted cadlag R`−valued process. Then

µX(ω; dt,dx) :=∑s>0

1[∆X 6=0](ω, s)δ(s,∆Xs(ω))(dt,dx)16

defines an integer-valued G−random measure on R× R`.

Proof. See Jacod and Shiryaev [41, Proposition II.1.16].

Actually, every integer-valued measure µ has an analogous representation. More precisely, [41, Propo-sition II1.14] verifies that for every integer-valued random measure µ there exists a thin, G−optional setD and an E−valued optional process β such that

µ(ω; dt,dx) =∑s≥0

1D(ω, s)δ(s,βs(ω))(dt,dx).

Notation I.78. We will maintain the notation of the last proposition and we will denote the integer-valued G−random measure associated to a G−adapted and cadlag process X by µX .

Now that we have defined the necessary objects, we can start presenting the desirable machinery.We start with the core of this section. The following theorem allows us to transform the process W ∗ µ,where W is a G−optional and µ an integer-valued G−random measure, into a G−martingale.

16Recall that δz denotes the Dirac measure sitting at z.


Theorem I.79. Let µ ∈ Aσ(G). There exists a random measure, called the compensator of µ under Gand denoted by νG, which is unique up to a P−null set, and which is characterised as being a G−predictablerandom measure satisfying either of the two following equivalent properties:

(i) For every non-negative G−predictable function W holds E[W ∗ µ∞] = E[W ∗ νG∞].(ii) If W is a G−predictable function such that |W | ∗ µ ∈ A+(G;R), then |W | ∗ νG ∈ A+(G;R) and

the process W ∗ µ − W ∗ νG is a uniformly integrable G−martingale. In other words, W ∗ νG is thecompensator under the filtration G of the process W ∗ µ.Moreover, there is a version of νG such that νG(s × R`) ≤ 1 identically.

Proof. See Jacod and Shiryaev [41, Theorem II.1.8] and [41, Proposition II.1.17].

Remark I.80. (i) For every a random measure µ ∈ Aσ(G) we are going hereinafter to use the “nice”version of νG for which the property νG(s × R`) ≤ 1 holds for every s ∈ R+.

(ii) Observe that for every cadlag process X holds 1R+×0 ∗ µX = 0 P−almost surely. This, in view

of Theorem I.79, in particular implies that∫

(0,∞)×0 ν(X,G)(ds,dx) = 0 P−almost surely.

When it is clear to which filtration refers the measurability of an integer-valued random measure,then we will not use the symbol of the filtration in the description. However, this is not the case for thecompensator of an integer-valued measure, where we will always prefer to denote the filtration. Sincein this dissertation we will be interested only for integer-valued measures associated to some adaptedprocess, we will present the rest results for these type of random measures. Therefore, we fix for the restof the section an arbitrary R`−valued, G−adapted and cadlag process X with associated integer-valuedmeasure µX .

Notation I.81. For the associated to X integer-valued random measure µX we adopt the followingnotation.

• The compensator of µX under the filtration G will be denoted by ν(X,G).• If X ∈ H2(G;R`), then the integer-valued measure associated to the jumps of the martingale X

will be denoted by µXd

. In other words, µXd

is the integer-valued measure associated to the purely

discontinuous part of the G−martingale X. In this case, µXd ∈ Aσ(G), i.e. its compensator is well-

defined. The compensator of µXd

under the filtration G will be denoted by ν(Xd,G). The reason for thischange in notation will be discussed at the end of the section.• The G−compensated integer-valued measure µ(X,G) is defined by

µ(X,G)(ω; dt, dx) := µX(ω; dt,dx)− ν(X,G)(ω; dt,dx).

• For W an E−valued G−predictable function and a G−predictable time ρ we denote∫R`W (ρ, x)µX(ρ × dx)(ω) := W (ω, ρ(ω),∆Xρ(ω)(ω))1[∆Xρ 6=0,‖W (ω,ρ(ω),∆Xρ(ω)(ω))‖1<∞]

+∞1[∆Xρ 6=0,‖W (ω,ρ(ω),∆Xρ(ω)(ω))‖1=∞]

and ∫R`W (ρ, x)ν(X,G)(ρ × dx)(ω) :=

∫R+×R`

W (ω, ρ(ω), x)1JρK(ω, t)ν(X,G)(ω; dt,dx),

if∥∥ ∫

R+×R`W (ω, ρ(ω), x)1JρK(ω, t)ν(X,G)(ω; dt,dx)

∥∥1< ∞, otherwise we define it to be equal to ∞.

Moreover,∫R`W (ρ, x)µ(X,G)(ρ × dx)(ω)

:=

∫R`W (ρ, x)µX(ρ × dx)(ω)−

∫R+×R`

W (ω, ρ(ω), x)1JρK(ω, t)ν(X,G)(ω; dt,dx).

The following remark justifies the forthcoming Definition I.83.

Remark I.82. By Jacod and Shiryaev [41, Lemma II.1.25] we have for every G−predictable time ρ that∫R`W (ρ, x)ν(X,G)(ρ × dx) = E

[ ∫R`W (ρ, x)µX(ρ × dx)

∣∣∣∣Gρ−] on [ρ <∞].


In other words, for every G−predictable time ρ

E[ ∫

R`W (ρ, x)µ(X,G)(ρ × dx)

∣∣∣∣Gρ−] = 0 on [ρ <∞].

If, moreover, for every G−predictable time ρ

E[∥∥∥∥ ∫

R`W (ρ, x)µ(X,G)(ρ × dx)

∥∥∥∥1

1[ρ<∞]

]<∞,

then the two above properties imply by Theorem I.21 that

ΠGp

(∫R`W (·, x)µ(X,G)(· × dx)

)= 0,

i.e. its G−predictable projection is indistinguishable from the zero process. Now, by [41, Theorem I.4.56a)], to every G−optional process H with the following properties

(i) H0 = 0,(ii) ΠG

p (H) = 0 and

(iii)∑s≤· ‖Hs‖22 ∈ A+(G;R)

we can associate a Y ∈ H2(G;E) such that ∆Y = H. Recalling Corollary I.9 we realise that we canactually associate a unique, up to indistinguishability, element of H2,d(G;E), namely Y d.

Definition I.83. Let µX ∈ Aσ(G). An E−valued G−predictable function W is said to be stochasticallyintegrable with respect to µ(X,G) if W ∈ H2(G, µX ;E), where

H2(G, µX ;E) :=

U :

(Ω, PG)→ (

E,B(E)),∑s≤·

∥∥∥∥∫R`W (s, x)µ(X,G)(s × dx)

∥∥∥∥2

2

∈ A+(G;R)

.

In this case, the stochastic integral of W with respect to µ(X,G) is defined to be the element of H2,d(G;E),which will be denoted by W ? µ(X,G), such that

∆(W ? µ(X,G))· =

∫R`W (·, x)µ(X,G)(· × dx)

up to indistinguishability.

Notation I.84. To the class H2(G, µX ;E) we associate the space K2(G, µX ;E) where

K2(G, µX ;E) :=W ? µ(X,G),W ∈ H2(G, µX ;E)

. (I.19)

In other words, K2(G, µX ;E) is the space of the E−valued stochastic integrals with respect to the pair(G, µX).

Proposition I.85. Let W be a G−predictable function such that |W | ∗ µX ∈ A+(G;R) or equivalently|W | ∗ ν(X,G) ∈ A+(G;R). Then the stochastic integral W ? µ(X,G) is well-defined. Moreover, the followingare true

(i) W ? µ(X,G) is a G−martingale of finite variation.(ii) W ? µ(X,G) = W ∗ µX −W ∗ ν(X,G).

Proof. See Jacod and Shiryaev [41, Proposition II.1.28].

Proposition I.86. Let H be a bounded G−predictable process and W ∈ H2(G, µX ;E). Then HW ∈H2(G, µX ;E) and H ·

(W ? µ(X,G)

)= (HW ) ? µ(X,G).

Proof. See [41, Proposition II.1.30].

Proposition I.87. X is G−quasi-left-continuous if and only if there exists a version of ν(X,G) that

satisfies

∫R`ν(X,G)(t × dx) = 0 for every (ω, t) ∈ Ω× R+.

Proof. See Jacod and Shiryaev [41, Corollary II.1.19].


Notation I.88. For every Γ :(Ω,G ⊗ B(R+)⊗ B(R`)

)→(E,B(E)

)we define

MµX [Γ] := E[Γ ∗ µX∞

].

We will refer to MµX as the Doleans-Dade measure associated to the random measure µX . We have abusedthe terminology for MµX , however, when we restrict the domain of MµX on the indicator functions of

PG−measurable sets, then it is easy to check that MµX is a σ−finite measure on(Ω, PG). We will denote

by MµX |PG the restriction of MµX on the set of G−predictable functions.

Definition I.89. The integer-valued measure µX is said to be a G−martingale (integer-valued random)measure if MµX |PG = 0.

Theorem I.90. Let µ be an integer-valued random measure. The following are equivalent:

(i) µ is a G−martingale integer-valued random measure.(ii) νG = 0.

(iii) For every E−valued G−predictable function W such that there exists an increasing sequence of

G−stopping times, say (τk)k∈N, for which limk→∞ τk = ∞ P−almost surely and (W ∗ µ)τk ∈ A(G;E)

for every k ∈ N, the process (W ∗ µ)τk ∈M(G;E) for every k ∈ N.


Corollary I.91. The integer-valued random measure µ(X,G) is a G−martingale measure.

Proof. Using Theorem I.79 we can see that the compensator of µ(X,G) is 0. Now, we can concludeby Theorem I.90.

Definition I.92. Let W :(Ω,G ⊗ B(R+) ⊗ B(R`)

)−→

(Rp,B(Rp)

)be such that ‖W‖1µX ∈ Aσ(G).

Then we define the PG−conditional projection of W on µX , denoted by MµX[W |PG], as the Rp−valued

G−predictable function with

MµX[W |PG]i :=

dMW iµX |PG

dMµX |PGfor every i = 1, . . . , p.

Theorem I.93. Let W ∈ H2(G, µX ;R) and N be a uniformly integrable real-valued G−martingale such

that |∆N |µ ∈ Aσ(G). Then, if [W ? µ(X,G), N ] ∈ A(G;R), we have

〈W ? µ(X,G), N〉· = (WMµ

[∆N

∣∣PG]) ∗ ν(X,G)· .


Theorem I.94. Let W ∈ H2(G, µXd ;Rp). Then

〈U ? µ(Xd,G)〉· =

∫(0,·]×R`

(U(s, x)

)>U(s, x) ν(Xd,G)(ds,dx)

−∑s≤·

(∫R`U(s, x) ν(s × dx)

)> ∫R`U(s, x) ν(s × dx).


The purpose of the following discussion is to clarify a subtle point in the construction of the stochasticintegral with respect to µY , where Y is an R`-valued, G−adapted and not G−quasi-left-continuous

process. Assume additionally that Tr[q] ∗ µY ∈ A+(G;R).17 Then µY ∈ Aσ(G) and its compensatorν(Y,G) are well-defined. Since we have assumed that Y is not G−quasi-left-continuous, there exists afinite G−stopping time ρ such that ν(Y,G)(ρ × R`) > 0 P−almost surely, for every version of ν(Y,G).This assumption can be justified by Proposition I.87.

Let us proceed now with the observation that Id` ∈ H2(G, µY ;R). Therefore, we can associate to theprocess Y the G−martingale Id` ? µ

(Y,G) ∈ H2,d(G;R`). Abusing notation we will denote Id` ? µ(Y,G) by

Y d, i.e

∆Y dt :=

∫R`x µ(Y,G)(t × dx) = ∆Yt −

∫R`x ν(Y,G)(t × dx). (I.20)

17The function q has been defined on p. 2.


At the end of the current discussion we will see that, if Y ∈ H2(G;R`), then its purely discontinuouspart and Id` ? µ

(Y,G) are indistinguishable, which does not lead to a notational conflict. Now that we

have constructed the square-integrable G−martingale Y d, we have also the option to work with µYd

with

associated compensator ν(Y d,G). Then, we can write (I.20) as∫R`x µ(Y,G)(t × dx) = ∆Y d =

∫R`xµY

d

(t × dx). (I.21)

A seemingly naive question that one may pose is whether µ(Y,G) and µ(Y d,G) generate the same spaces of

stochastic integrals. Of course, we expect that the compensators ν(Y,G) and ν(Y d,G) do not coincide, since

they correspond to different integer-valued measures µY and µYd

. Therefore, in general K2(G, µX ;E) 6=K2(G, µXd ;E) for the arbitrary state space E. From a different perspective, we know by Corollary I.91

that µ(Y,G) is a G−martingale measure, while for µYd

this is not true. For example, if V is a positive

real-valued G−predictable function such that V ∗ µY d ∈ A+(G;R), then we obtain a G−submartingale.So, we proceed by examining a G−predictable function W whose value W (ω, s, x) is not linear on x.

Since an analogous function will be used in Chapter III in order to construct suitable martingale sequences,it is a good opportunity to see in detail why we will finally prefer the G−compensated integer-valued

measure µ(Y d,G). This will justify also the second bullet in Notation I.81, where we substitute the notation

of the integer-valued measure associated to the process X with µXd

, if the process X ∈ H2(G;R`).Say W (ω, s, x) = ‖x‖22 ∧ 1.18 Then, the jump of W ? µ(Y,G) at ρ, which was fixed at the beginning of

the discussion, is given by∫R`

(‖x‖22 ∧ 1

)µY (ρ × dx)−

∫R`

(‖x‖22 ∧ 1

)ν(Y,G)(ρ × dx) =

∥∥∆Yρ∥∥2

2∧ 1− E

[∥∥∆Yρ∥∥2

2∧ 1∣∣∣Gρ−],

while the respective jump of W ? µ(Y d,G) is given by∫R`

(‖x‖22 ∧ 1

)µY

d

(ρ × dx)−∫R`

(‖x‖22 ∧ 1

)ν(Y d,G)(ρ × dx) =

∥∥∆Y dρ∥∥2

2∧ 1− E

[∥∥∆Y dρ∥∥2

2∧ 1∣∣∣Gρ−]

(I.21)=

∥∥∥∥∫R`x µ(Y,G)(ρ × dx)

∥∥∥∥2

2

∧ 1− E[∥∥∥∥ ∫

R`x µ(Y,G)(ρ × dx)

∥∥∥∥2

2

∧ 1

∣∣∣∣Gρ−]=

∥∥∥∥∆Yρ −∫R`x ν(Y,G)(ρ × dx)

∥∥∥∥2

2

∧ 1− E[∥∥∥∥∆Yρ −

∫R`x ν(Y,G)(ρ × dx)

∥∥∥∥2

2

∧ 1

∣∣∣∣Gρ−].It is clear now, that the spaces K2(G, µY ;E) and K2(G, µY d ;E) are not equal. However, when Y ∈H2(G;R`), then we have by Jacod and Shiryaev [41, Lemma I.2.27] that ΠG

p (Y ) = Y−. In other words,∫R`x ν(Y,G)(ω; t × dx) = 0 for every (ω, t)× Ω× R+.

Therefore, the purely discontinuous part of the G−martingale Y has jumps ∆Y , which in view of theabove property implies that

∆Yt(ω) =

∫R`xµY (ω; t × dx).

This verifies our comment at the beginning of the discussion, that the abuse of notation (I.20) does notconflict with the case Y ∈ H2(G;R`). Intuitively, one can say that having already a G−martingale, welose the information of the jump-compensating procedure which transforms a G−adapted process to aG−martingale.

The above discussion was made on the basis that Y is a G−adapted process which is not G−quasi-left-continuous. In view of Definition I.15 and Proposition I.87, this subtle difference vanishes when theprocess Y is G−quasi-left-continuous, hence the compensator ν(Y,G) has no jumps.

Henceforth, assuming that X is a G−martingale, when we refer to the jump process ∆Xd we willmean the R`−valued process

∆Xd· :=

(∫R`π1(x)µ(X,G)(· × dx), . . . ,

∫R`π`(x)µ(X,G)(· × dx)

), (I.22)

18The fact that W ∈ H2(G, µY d ;R) is an outcome of the definition of Y d and that W ∈ H2(G, µY ;R).


while, assuming that X is simply an G−adapted process, when we refer to the jump process ∆X we willmean the R`−valued process

∆X· =

(∫R`π1(x)µX(· × dx), . . . ,

∫R`π`(x)µX(· × dx)

). (I.23)

Theorem I.95. Let Y ∈ H2(G,Rp) and assume that X ∈ H2(G;R`). Then, there exists unique, up to

indistinguishability, U ∈ H2(G, µXd ;Rp) and N ∈ H2(G;Rp) with MµXd[∆N

∣∣PG] = 0 such that

Y = U ? µ(Xd,G) +N.

The G−predictable function U is uniquely determined up to indistinguishability by the triplet(G, µXd , Y

).

If, moreover, X is G−quasi-left-continuous, then U is indistinguishable from MµXd[∆Y

∣∣PG].Proof. See Jacod and Shiryaev [41, Theorem III.4.20].

I.5. Selected results from stochastic calculus

In this section we collect some well-known results from stochastic calculus which we are going to useheavily. However, the Burkholder–Davis–Gundy Inequality and the Doob’s LΥ−Inequality will be statedwith the help of Young functions, which are not the forms that are usually met in the literature. We willrely on the generality of these results in order to prove Theorem III.3 by utilising Proposition I.51.

For the current section we work in the G−stochastic basis, where G is an arbitrary filtration.

Definition I.96. Let X ∈ M(G;Rp) and Y ∈ M(G;Rq). We define the (optional) quadratic variationof the pair (X,Y ) is the Rp×p−valued G−optional process [X,Y ] which is defined for every t ∈ R+ by

[X,Y ]ijt := 〈Xc,i, Y c,j〉t +∑

0<s≤t

∆Xis∆Y

js for i = 1, . . . , p and j = 1, . . . , q.

Remark I.97. For X,Y ∈M(G;Rp), the quadratic variation satisfies the so called polarisation identity

[X,Y ] =1

4

([X + Y ]− [X − Y ]

).

Proposition I.98. Let S be a G−adapted process such that it can be written in the form S = M + A,where M ∈M(G;R) and A ∈ V(G;R). Then, the following are equivalent:

(i) S ∈ Ssp(G;R).(ii) supu≤· |Su − S0| ∈ A+(G;R).


Proposition I.99. Let X,Y ∈ H2(G;R). Then [X,Y ] ∈ A(G;R) and [X,Y ]− 〈X,Y 〉 ∈ M(G;R).


Theorem I.100 (Kunita–Watanabe Inequality). Let X,Y ∈ M(G;R) and H,K be B ⊗ G−measurableprocesses. Then for every t ∈ R+

19 holds∫(0,·]|Hs| |Ks|dVar

([X,Y ]

)s≤(∫

(0,·]|Hs|2 d[X]s

) 12(∫

(0,·]|Ks|2 d[Y ]s

) 12 P− a.s. (I.24)

and if 〈X〉, 〈Y 〉 and 〈X,Y 〉 all exist,∫(0,·]|Hs| |Ks|dVar

(〈X,Y 〉

)s≤(∫

(0,·]|Hs|2 d〈X〉s

) 12(∫

(0,·]|Ks|2 d〈Y 〉s

) 12 P− a.s.. (I.25)

Moreover,

E[ ∫

(0,·]|Hs| |Ks|dVar

([X,Y ]

)s

]≤ E

[ ∫(0,·]|Hs|2 d[X]s

] 12

E[ ∫

(0,·]|Ks|2 d[Y ]s

] 12

(I.26)

and if 〈X〉, 〈Y 〉 and 〈X,Y 〉 all exist,

E[ ∫

(0,·]|Hs| |Ks|dVar

(〈X,Y 〉

)s

]≤ E

[ ∫(0,·]|Hs|2 d〈X〉s

] 12

E[ ∫

(0,·]|Ks|2 d〈Y 〉s

] 12

. (I.27)

19In the case t =∞ the domain of the Lebesgue–Stieltjes integral is written (0,∞).

I.5. SELECTED RESULTS FROM STOCHASTIC CALCULUS 27


Theorem I.101 (Burkholder–Davis–Gundy Φ−Inequality). If X is a real-valued G−martingale andΦ ∈ YFmod, then

1

4scΦ‖ supt∈R+

|X|‖Φ ≤ ‖[X]12∞‖Φ ≤ 6scΦ‖ sup

t∈R+

|X|‖Φ.

If at least one of the quantities ‖ supt∈R+|X|‖Φ, ‖[X]

12∞‖Φ, E[Φ(supt∈R+

|X|)] and E[Φ([X]12∞)] is finite,

then there exist positive constants cΦ, CΦ (depending only on Φ) such that

cΦE[Φ( supt∈R+

|X|)] ≤ E[Φ([X]12∞)] ≤ CΦE[Φ( sup

t∈R+

|X|)].

Proof. For the first inequality see Dellacherie and Meyer [27, Theorem VII.92, p. 287]. Observethat we can obtain the Burkholder–Davis–Gundy Inequality in a slightly more general framework, i.e.when Φ is a convex and moderate function such that Φ(0) = 0.

For the second inequality, since Φ is moderate we have from Proposition I.53.(v) that E[Φ([X]12∞)] < ∞

if and only if ‖[X]12∞‖Φ < ∞ and analogously E[Φ(supt∈R+

|X|)] < ∞ if and only if ‖ supt∈R+|X|‖Φ < ∞.

Then, for the constants cΦ, CΦ of second inequality see He et al. [35, Theorem 10.36].

We state also the Burkholder–Davis–Gundy Inequality in its mostly-known form because it will allowus to identify the space

H1(G;R) :=M ∈ M(G;R), sup

t∈R+

∣∣Mt

∣∣ ∈ L1(G;R)with the space

M ∈ M(G;R), [M ]

12∞ ∈ L1(G;R). Obviously, H1(G;R) ⊂ M(G;R). Moreover, the

inclusion is strict; solve Revuz and Yor [60, Exercise 3.15 of Chapter II].

Lemma I.102. Let Φ ∈ YF and X,Y ∈ M(G;R) such that E[Φ([X]

12∞)], E[Φ([Y ]

12∞)]

< ∞. Then

[X,Y ] ∈ A(G;R) and, consequently, its compensator [X,Y ]p,G is well-defined.

Proof. The integrability conditions E[Φ([X]

12∞)], E[Φ([Y ]

12∞)]

< ∞ verify that [X]12∞, [Y ]

12∞ are fi-

nite P−almost surely. By Kunita–Watanabe Inequality (I.24) and Young’s Inequality, see Lemma I.44.(i),we have

Var([X,Y ]

)∞ ≤ [X]

12∞[X]

12∞ ≤ Φ

([X]

12∞)+Φ

([Y ]

12∞).

Then, we can conclude the existence of [X,Y ]p,G by Theorem I.64.

Theorem I.103 (Burkholder–Davis–Gundy Lϑ−Inequality). If X is a real-valued G−martingale withM0 = 0 and ϑ ≥ 1, then there exist constants cϑ, Cϑ such that

cϑE[( supt∈R+

|X|)ϑ] 1

ϑ ≤ E

[[X]

ϑ2∞] 1

ϑ ≤ scϑE[( supt∈R+

|X|)ϑ] 1

ϑ

.


We conclude this section with the celebrated Doob’s Inequality and a useful lemma whose mainargument relies on Doob’s LΥ−Inequality. The latter will be useful in Chapter IV.

Theorem I.104 (Doob’s LΥ−Inequality). Let Υ ∈ YF such that Υ ∈ YFmod and X be

(i) either a real-valued G−martingale(ii) or a real-valued positive cadlag G−submartingale.

Then

‖ supt∈R+

|Xt|‖Υ ≤ 2scΥ ‖X∞‖Υ.

Proof. If X is a positive cadlag G−submartingale see Dellacherie and Meyer [27, Paragraph VI.103,p. 169] and [27, Paragraph VI.97 p. 161],[27, Remark VI.21, p. 82] for the notation. IfX is aG−martingale,20

then by Jensen’s Inequality |X| is a positive cadlag G−submartingale, which reduces to the formercase.

20Recall that we use its cadlag modification


The classical form of Doob’s Inequality is presented in the next corollary. Observe however that theconstant of Corollary I.105 is improved comparing to the one we get from Theorem I.104. The interestedreader may have the answer in Rao and Ren [58, Section 2.4].

Corollary I.105 (Doob’s Lϑ−Inequality). Let ϑ, ϑ ∈ (1,∞) be such that 1ϑ + 1

ϑ = 1 holds and X be

(i) either a real-valued G−martingale(ii) or a real-valued positive cadlag G−submartingale.

Then (E

[supt∈R+

|Xt|ϑ]) 1

ϑ ≤ ϑ(E[ |X∞|ϑ ]) 1

ϑ

.

Proof. See Dellacherie and Meyer [27, Paragraph VI.101, p. 166]. Lemma I.106. Let (Gk)k∈N

be a sequence of filtrations, where Gk :=(Gk

t

)t∈R+

, and (ξk)k∈Nbe a

sequence of Rp−valued random variables such that(∥∥ξk∥∥2

2

)k∈N

is uniformly integrable. Then the sequence(supt∈R+

∥∥E[ξk∣∣Gkt

]∥∥22

)is uniformly integrable.

Proof. Let Φ ∈ YFmod be a Young function for which the sequence(∥∥ξk∥∥2

2

)k∈N

satisfies the de LaVallee Poussin–Meyer Criterion, i.e.

supk∈N

E

[Φ(∥∥ξk∥∥2

2

)]< ∞. (I.28)

Then Ψ := Φquad ∈ YFmod (see Proposition I.51.(i)). Using the fact that Φ is increasing, we can write(I.28) as

M := supk∈N

E

[Ψ(‖ξk‖2)] < ∞. (I.29)

The latter form will be more convenient for later use. Before we proceed to prove the claim of the lemma,we provide some helpful results. In order to ease notation, let us denote the Orlicz norm

∥∥‖ξk‖2∥∥Ψ by

Ξk for k ∈ N. Observe that Ξk < ∞, for every k ∈ N because of (I.29). We are going to prove thatsupk∈N

Ξk < ∞. To this end, observe that

If M ≤ 1, then supk∈NΞk ≤ 1 by the definition of the Orlicz norm; see Notation I.52.

If M > 1, then using the convexity of Ψ and the fact that Ψ(0) = 0 we obtain

E

[Ψ(‖ξk‖2

M

)]≤ E

[Ψ(‖ξk‖2)]M

(I.29)

≤ 1 which implies Ξk ≤ M.

and consequently

supk∈N

Ξk ≤ 1 ∨M(I.29)< ∞. (I.30)

We proceed now to prove uniform integrability of(supt∈R+


]∥∥22

)k∈N

. By de La Vallee Poussin

Theorem, see Theorem I.27, it suffices to prove that supk∈NE

[Φ(supt∈R+

∥∥E[ξk|Gkt ]∥∥22

)]< ∞, or equiv-

alently that supk∈NE

[Ψ(√

2Sk)]

< ∞, where we have defined Sk := supt∈R+

∥∥E[ξk|Gkt ]∥∥2. Analogously

to (I.14) we can obtain

E

[Ψ(√

2Sk)] ≤ 1 ∨ sc‖

√2Sk‖Ψ

Ψ. (I.31)

Recall that Ψ ∈ YFmod due to its definition Ψ = Φ quad. By Proposition I.51 we have that alsoΨ ∈ YFmod. Therefore, sc

Ψ, sc

Ψ = cΨ

∈ (1,∞). For the validity of the last argument recall Notation I.41,Definition I.43 and Corollary I.45. Consequently, we can conclude the required property if we prove thatsupk∈N

‖Sk‖Ψ < ∞, where the√2 can be taken out, since ‖ · ‖Ψ is a norm. To this end we will utilise

Doob’s LΥ−inequality. We can obtain now by standard properties of norms and the fact that ‖·‖2 ≤ ‖·‖1(and consequently Sk ≤ supt∈R+

∥∥E[ξk|Gkt ]∥∥1) the following inequalities∥∥Sk

∥∥Ψ

≤∥∥∥ supt∈R+

∥∥E[ξk|Gkt ]∥∥1

∥∥∥Ψ

≤p∑

j=1

∥∥∥ supt∈R+

∣∣E[ξk,j |Gkt ]∣∣∥∥∥

Ψ

Doob’s In.≤ 2scΨ

p∑j=1

∥∥ξk,i∥∥Ψ

≤ 2pscΨ

∥∥‖ξk‖2∥∥Ψ = 2pscΨΞ

k.

I.6. THE SKOROKHOD SPACE 29

The above inequality and (I.30) imply the desired

supk∈N

∥∥Sk∥∥Ψ

≤ 2pscΨ sup

k∈N

Ξk < ∞,

which, in view of Equation (I.31), implies also the finiteness of supk∈NE[Ψ(

√2Sk)

]. Therefore, the

sequence(supt∈R+


]∥∥22

)is uniformly integrable since it satisfies the de La Vallee Poussin theorem

for the Young function Φ.

I.6. The Skorokhod space

In this section we present the Skorokhod space D(E) which is the natural path-space for semi-martingales adapted to some filtration that satisfies the usual conditions. The weaker of the topologiesthe Skorokhod space will be endowed with, will be the (Skorokhod) J1−topology. The J1−topology ismetrisable and makes the space to be complete and separable, i.e. D(E) becomes Polish. The mainreference of this section is Jacod and Shiryaev [41].

Definition I.107. (i) The function α : R+ −−→ E will be called cadlag if α is right-continuous andfor every t ∈ R the value α(t−) exists.

(ii) The E−Skorokhod space is the space of all cadlag, E−valued functions defined on R+ and it willbe denoted by D(E), i.e.

D(E) := α : R+ −−→ E,α is cadlag.The zero function will be denoted by 0.(iii) The filtration D(E) :=

(Dt(E)

)t∈R+

is the right-continuous filtration obtained by the filtration

generated by the canonical projections, i.e.

Dt(E) :=⋂u>t

σ(α(s), α ∈ D(E) and s ≤ u)

Moreover, D(E) := σ(α(s), α ∈ D(E) and s ∈ R+).

We will mostly denote the elements of D(R) with small Greek letters, usually the first three one ofthe Greek alphabet, i.e. α, β and γ.

Definition I.108. (i) Let λ : R+ −−→ R+ be continuous and increasing such that λ(0) = 0 andlimt→∞ λ(t) = ∞. Then λ will be called time change.

(ii) The set of all time changes will be denoted by Λ, i.e.

Λ := λ : R+ −−→ R+, λ is continuous and increasing with λ(0) = 0, limt→∞λ(t) = ∞.

(iii) To each α ∈ D(E) and each interval I ⊂ R+ we associate the moduli

w(α; I) := sups,t∈I

‖α(s)− α(t)‖2

and

w′N (α, θ) := inf

maxi≤r

w(α; [ti−1, ti)), 0 = t0 < t1 < · · · < tr = N with infi<r

(ti − ti−1) ≥ θ.

Theorem I.109. There is a metrisable topology on D(E), called the Skorokhod J1−topology, for whichthe space is Polish. Moreover, the following are equivalent:

(i) The sequence (αk)k∈N converges to α∞ under the J1−topology.(ii) There exists a sequence (λk)k∈N ⊂ Λ such that

sups∈R+

|λk(s)− s| −−−−→k→∞

0 and sups≤N

‖αk(λk(s))− α∞(s)‖2 −−−−→k→∞

0, for all N ∈ N. (I.32)

The metric under which D(E) becomes Polish will be denoted by δJ1(E).21 Moreover, the associated Borel

σ−algebra equals the σ−algebra D(E), which was defined Definition I.107.(ii).

21We will omit the state space from the notation, when it is clear to which one we refer.


Proof. See Jacod and Shiryaev [41, Theorem VI.1.14 a)]. For the definition of δJ1see [41, Definition

VI.1.26]. In [41, Lemma VI.1.30, Lemma VI.1.31, Corollary VI.1.32] it is proven that δJ1 is indeed ametric. In [41, Lemma VI.1.33] it is proven that (D(E), δJ1) is complete. The separability of the space isproven in [41, Corollary VI.1.43 a)].

For (i) ⇒ (ii) see [41, Lemma VI.1.31], while for the implication (ii) ⇒ (i) see [41, Lemma VI.1.44].For the validity of the statement regarding the Borel σ−algebra and D(E) see [41, Lemma VI.1.45,Lemma VI.1.46].

Remark I.110. Recall the well-known fact that a cadlag function α has at most countably many pointsof discontinuities; see Billingsley [7, Section 12, Lemma 1] for the case the domain is [0, 1] and then onecan easily conclude for the case R+, as a countable union of finite sets is a countable set. Therefore, thefunction ∆α is zero except from a set of Lebesgue measure zero.

Theorem I.111. A set D ⊂ D(E) is relatively compact for the Skorokhod J1−topology if and only iffor every N ∈ N both of the following conditions hold:

supα∈D

sups≤N‖α(s)‖2 <∞ and lim

θ↓0supα∈D

w′N (α, θ) = 0 for every N ∈ N.

Proof. See Jacod and Shiryaev [41, Theorem VI.1.14 b)]. More precisely, [41, Lemma VI.1.39]proves the necessity and [41, Corollary VI.1.43] provides the sufficiency.

We will say that the sequence (αk)k∈N is a δJ1(E)−convergent sequence if it satisfies Theorem I.109

and we will denote its convergence also by αkJ1(E)−−−−−→ α∞. Apart from the J1−topology we will equip

the space D(E) with two more topologies, which are presented below.

Notation I.112. Let α, β ∈ D(E).

• The locally uniform topology is the one associated with the metric δlu, where

δlu(α, β) =∑k∈N

1

2k(w(α− β; [0, k] ∧ 1

).

The convergence of a sequence (αk)k∈N to α∞ under the metric δlu will be denoted by αklu−−→ α∞.

• The uniform topology is associated with the norm ‖ · ‖∞, where‖α‖∞ := sups∈R+‖α(s)‖2. The

convergence of a sequence (αk)k∈N to α∞ under the metric δ‖·‖∞ will be denoted by αk‖·‖∞−−−−→ α∞.

The next proposition provides the relationship between δJ1(D(E)) and δlu.

Proposition I.113. (i) The J1−topology is weaker than the locally uniform topology.(ii) Let α∞ ∈ D(E) be a continuous function. Then

αkJ1(D(E))−−−−−−−→ α∞ if and only if αk

δlu−−−→ α∞.

Proof. See Jacod and Shiryaev [41, Proposition VI.1.17].

Remark I.114. For the metrics δJ1(D(E)), δlu and δ‖·‖∞ holds

δJ1(D(E))(α, 0) = δlu(α, 0) for every α ∈ D(E)

and

δJ1(D(E))(α, β) ≤ δlu(α, β) ≤ sups∈R+

‖α(s)− β(s)‖2 for every α, β ∈ D(E).

The following lemma is essentially [35, Example 15.11]. We provide an alternative proof in somespecial cases of the aforementioned result since we will need later to elaborate on similar cases.

Lemma I.115. Let (E, ‖ · ‖2), (xki )k∈N ⊂ E and (tki )k∈N ⊂ R+, for i = 1, 2, such that tk1 < tk2 for every

k ∈ N.

(i) If tk1|·|−−→∞, then xk11[tk1 ,∞)

J1(E)−−−−−→ 0.

(ii) If xk1‖·‖2−−−→ 0, then xk11[tk1 ,∞)

J1(E)−−−−−→ 0.

(iii) If tk1|·|−−→ t∞1 ∈ R+ and xk1

‖·‖2−−−→ x∞1 , then xk11[tk1 ,∞)

J1(E)−−−−−→ x∞1 1[t∞1 ,∞).


(iv) If tk1|·|−−→ t∞1 ∈ R+, tk2

|·|−−→∞ and xk1‖·‖2−−−→ x∞1 , then

xk11[tk1 ,∞) + xk21[tk2 ,∞)

J1(E)−−−−−→ x∞1 1[t∞1 ,∞).

(v) If tk1|·|−−→ t∞1 , tk2

|·|−−→ t∞2 , where t∞1 < t∞2 ∈ R+, xk1‖·‖2−−−→ x∞1 and xk2

‖·‖2−−−→ x∞2 then

xk11[tk1 ,∞) + xk21[tk2 ,∞)

J1(E)−−−−−→ x∞1 1[t∞1 ,∞) + x∞2 1[t∞2 ,∞).

Proof. For (i) and (ii), since the limit function is (trivially) continuous, we can use Proposition I.113in order to justify that we can choose without loss of generality λk = Id for every k ∈ N. In view ofTheorem I.109 it is therefore only left to prove that

sups≤N‖αk(s)‖2 −−→ 0 for every N ∈ N.

(i) Assume that tk1|·|−−→ ∞. Then for every N ∈ N there exists a k0(N) ∈ N such that tk1 ≥ N for

every k ≥ k0(N). Therefore, for every fixed N ∈ N, it holds sups≤N ‖αk(s)‖2 = 0 for every k ≥ k0(N).

(ii) Assume now xk1‖·‖1−−−→ 0, i.e. for every ε > 0 there exists a k0(ε) ∈ N such that ‖xk‖2 < ε. Let

us fix an ε > 0 and an N ∈ N, then sups≤N ‖xk‖2 < ε for every k ≥ k0(ε), which allows us to conclude.Observe that the choice of k0 is independent of N.

For (iii)-(v) see He et al. [35, Example 15.11], which deals with convergence of piecewise constantfunctions with possibly infinitely many jumps. For this reason recall conditions (i)-(iii) of He et al. [35,Remark 15.32] for the detailed properties of the jump times.

Remark I.116. The alert reader may have observed the condition imposed in Lemma I.115.(v) for thejump-times t∞1 and t∞2 , i.e. t∞1 < t∞2 . Let us examine what may fail in the case t∞1 = t∞2 . To this endwe set the following framework: assume that tk1 ↑ t and tk2 ↓ t, i.e. tk1 → t with tk1 ≤ t and tk2 → t witht ≤ tk2 . Then, in view of Lemma I.115.(iii) we have that

αk1 := 1[tk1 ,∞)

J1(R)−−−−→ 1[t,∞) and αk2 := −1[tk2 ,∞)

J1(R)−−−−→ −1[t,∞). (I.33)

On the other hand, the sequence (αk1 +αk2)k∈N does not converge under the J1−topology, since (1[tk1 ,tk2 ))k∈N

fails to be a δJ1(R)−Cauchy sequence. For the last claim see the comments right after Billingsley [7, Section12, Lemma 2, p. 126] in conjunction with [7, Example 12.2].

Let us explain a bit more this (counter-)example. By Theorem I.109 we have that there exists asequence of time changes (λki )k∈N, for i = 1, 2, such that the respective convergence in (I.33) hold.However, the sum of the converging sequences is not converging, because we cannot find a sequence oftime changes (λk)k∈N such that it is finally common for the sequences (1[tk1 ,∞))k∈N, (1[tk2 ,∞))k∈N and such

that (I.33) holds.

After providing the main results for the Skorokhod J1−topology, we need to clarify some details.

Remark I.117. (i) The space (D(E), δJ1(E)) is not a topological vector space. This is direct byRemark I.116 which proves that the addition is not δJ1

−continuous.(ii) The space D(Rp×q) can be identified with D(R)p×q. Let us endow the space

D(R)p×q := α : R+ −−→ Rp×q, αij is cadlag for every 1 ≤ i ≤ p, 1 ≤ q.with the product topology, which is compatible with the metric

δΠ(α, β) :=

p∑i=1

q∑j=1

δJ1(R)(αij , βij).

However, the topological spaces (D(Rp×q), δJ1(Rp×q)) and (D(R)p×q, δΠ) do not coincide. Indeed, by

Remark I.116 we have

δΠ((αk1 , α

k2), (1[t,∞),1[t,∞))

)−−→ 0 but δJ1(Rp×q)

((αk1 , α

k2), (1[t,∞),1[t,∞))

)6−−→ 0.

Now it may be clear to the reader why we have always indicated the state space in the notation of themetric. We will (almost) always indicate the state space in the notation for the sake of clarity.

(iii) For notational convenience we have stated the main results for the space D(E) and not for thespace D(Rd), where d is a natural integer, as in Jacod and Shiryaev [41, Chapter VI]. This can be donewithout loss of generality by defining an isometry from every finite dimensional space (E, ‖ · ‖2) to thespace (Rp, ‖ · ‖2), where p denotes the dimension of the space E.


The following are classical criteria for obtaining the joint convergence.

Proposition I.118. Let (αk)k∈N ⊂ D(E). The convergence αkJ1(E)−−−−−→ α∞ holds if and only if whenever

(tk)k∈N ⊂ R+ be such that tk −−→ t∞ the following conditions hold:

(i)∥∥αk(tk)− α∞(t∞)

∥∥2∧∥∥αk(tk)− α∞(t∞−)

∥∥2−−→ 0.

(ii) If∥∥αk(tk) − α∞(t∞)

∥∥2−−→ 0 and (sk)k∈N ⊂ R+ be such that tk ≤ sk for every k ∈ N and

sk −−→ t∞, then∥∥αk(sk)− α∞(t∞)

∥∥2−−→ 0.

(iii) If∥∥αk(tk) − α∞(t∞−)

∥∥2−−→ 0 and (sk)k∈N ⊂ R+ be such that sk ≤ tk for every k ∈ N and

sk −−→ t∞, then∥∥αk(sk)− α∞(t∞−)

∥∥2−−→ 0.

Proof. See Ethier and Kurtz [33, Proposition 3.6.5].

Remark I.119. Assume αkJ1(E)−−−−−→ α∞ and tk −−→ t∞. Then by Proposition I.118.(i) we have that the

only possible limit points of the sequences(αk(tk)

)k∈N,

(αk(tk−)

)k∈N are α∞(t∞) and α∞(t∞−).

Proposition I.120. Let (αk)k∈N ⊂ D(E1) and (βk)k∈N ⊂ D(E2). The convergence (αk, βk)J1(E1×E2)−−−−−−−−→

(α∞, β∞) holds if and only if

(i) αkJ1(E1)−−−−−→ α∞,

(ii) βkJ1(E2)−−−−−→ β∞ and

(iii) for every t ∈ R+ there exists a sequence (tk)k∈N with tk −−→ t and such that ∆αk(tk)‖·‖2−−−→ ∆α∞(t)

and ∆βk(tk)‖·‖2−−−→ ∆β∞(t).

Proof. For the implication (i)⇒ (ii) apply Jacod and Shiryaev [41, Proposition VI.2.1 a)] and thenconclude using Remark I.117.(i). For the implication (ii) ⇒ (i) apply [41, Proposition VI.2.2 b)] for thespaces D(E1), D(E2) and then conclude using Remark I.117.(i).

Looking back to Remark I.116, it is clear that there exists no sequence (tk)k∈N that satisfies thecondition (iii) of Proposition I.120 for the point t.

The following proposition will be helpful to prove Corollary I.125.

Proposition I.121. Let (αk)k∈N be a δJ1(E)−convergent sequence and t ∈ R+.

(i) There exists a sequence (tk)k∈N such that tk −−→ t, αk(tk) −−→ α∞(t) and αk(tk−) −−→ α∞(t−).

Therefore, the convergence ∆αk(tk)‖·‖2−−−→ ∆α∞(t) also holds.

(ii) Let ∆α∞(t) 6= 0 and (tk)k∈N be a sequence such that tk −−→ t and ∆αk(tk) −−→ ∆α∞(t). Then,

any other sequence (sk)k∈N for which holds sk −−→ t and ∆αk(sk)‖·‖2−−−→ ∆α∞(t) coincides with (tk)k∈N

for k large enough.


The next classical result informs us that we can obtain the convergence of the sum of jointly convergentsequences.

Corollary I.122. Let((αk, βk)

)k∈N be a δJ1(E×E)−convergent sequence. Then

(αk, βk, αk + βk)J1(E×E×E)−−−−−−−−−→ (α∞, β∞, α∞ + β∞).

Proof. We will prove it for E = R. Let, therefore, (αk, βk)k∈N ⊂ D(R) and assume that

(αk, βk)J1(R2)−−−−−→ (α∞, β∞).

Choose a t ∈ R+ such that (∆α∞(t),∆β∞(t)) 6= 0. Then, by Proposition I.121 there exists a sequence(tk)k∈N such that tk −−→ t and

(∆αk(tk),∆βk(tk))‖·‖2−−−→ (∆α∞(t),∆β∞(t)).

In particular, we can conclude that ∆αk,1(tk) + ∆αk,2(tk)|·|−−→ ∆α∞,1(t) + ∆α∞,2(t). Now, we can

conclude by Proposition I.120 that

(αk, βk, αk + βk) −−→ (α∞, β∞, α∞ + β∞).


Remark I.123. The result above does not depend on the dimensions of the state spaces, and can begeneralized inductively to an arbitrary number of sequences, as long as a common converging sequenceexists.

A convenient variation of the Proposition I.121 is provided in Coquet, Memin, and S lominski [23,Lemma 1], which we state also here.

Lemma I.124. Let (αk)k∈N ⊂ D(Rp). The convergence αkJ1(Rp)−−−−−→ α∞ holds if and only if the following

hold

(i) αk,iJ1(R)−−−−→ α∞,i, for i = 1, . . . , p,

(ii)

q∑i=1

αk,iJ1(R)−−−−→

q∑i=1

α∞,i, for q = 1, . . . , p.

We can provide now a corollary which will be proven quite convenient, since it allows us to concludethe joint convergence of two sequences once we can find a common element between the elements of thetwo converging sequences.

Corollary I.125. Let (αk)k∈N, resp. (βk)k∈N, be a δJ1(Rp1 )−convergent, resp. δJ1(Rp2 )−convergent,

sequence. If there exist 1 ≤ i1 ≤ p1, 1 ≤ i2 ≤ p2 such that αk,i1 = βk,i2 for every k ∈ N, then

(αk, βk)J1(Rp1×Rp2 )−−−−−−−−−→ (α∞, β∞).

Proof. We will apply Proposition I.120. It is only left to verify that condition (iii) of the aforemen-tioned proposition holds. To this end let us fix a t ∈ R+. By (i), resp. (ii), and Proposition I.121 we havethat there exists a sequence (tk1)k∈N, resp. (tk2)k∈N, such that

∆αk(tk1) −−→ ∆α∞(t), resp. ∆βk(tk2) −−→ ∆β∞(t).

However, by (iii) and Proposition I.121.(ii) we have that (tk1)k∈N and (tk2)k∈N finally coincide. Therefore,we can choose one of the two time sequences, say (tk1)k∈N, in order to conclude

∆αk(tk1) −−→ ∆α∞(t), resp. ∆βk(tk1) −−→ ∆β∞(t),

Proposition I.120.(iii) is satisfied.

We close this part with another convenient result.

Lemma I.126. Let (αk)k∈N ⊂ D(E) and f : (E, ‖ · ‖2) −→ (Rp, ‖ · ‖2) be continuous. If αkJ1(E)−−−−→ α∞

then f(αk)J1(Rp)−−−−−→ f(α∞).

Proof. The continuity of f allows us to apply Proposition I.120.

I.6.1. J1−continuous functions. The aim of this sub–sub–section is the following: for a givenδJ1(E)−converging sequence (αk)k∈N and a given (not necessarily continuous) function g : E −→ R, weneed to determine suitable conditions for the function g and a set I ⊂ E such that∑

0<t≤·

g(∆αk(t))1I(∆αk(t))

J1(R)−−−−→∑

0<t≤·

g(∆α∞(t))1I(∆α∞(t)). (I.34)

This is Proposition I.134, which refines the classical result Jacod and Shiryaev [41, Corollary VI.2.8], andit will be crucial for constructing in Chapter III a family of J1−convergent sequences of submartingales. Inorder to obtain Corollary I.133, we need to refine another classical result, namely [41, Proposition VI.2.7];the refinement of the latter is Proposition I.132. For simplicity, we provide the results for E = Rq.

Before we proceed we introduce the necessary notation and provide a simple example, which willclarify to a great extend the main idea of Proposition I.132.

Definition I.127. For β ∈ D(Rq) and γ ∈ D(R) we introduce the sets

U(β) := u > 0,∃t > 0 with ‖∆β(t)‖2 = u,W (γ) := u ∈ R \ 0, ∃t > 0 with ∆γ(t) = u

and

I(γ) := (v, w) ⊂ R, v < w with vw > 0 and v, w /∈W (γ).


For α ∈ D(Rq) we define the set

J (α) := q∏i=1

Ii, where Ii ∈ I(αi) ∪R

for every i = 1, . . . , q\Rq.

The set W (γ), which is at most countable, collects the heights of the jumps of a real-valued functionγ. The set I(γ) collects all the open intervals of R \ 0 with boundary points of the same sign, which,moreover, do not belong to W (γ).

Remark I.128. For the rest of this subsection we will use additionally the notation αs for the value ofthe arbitrary α ∈ D(Rq) at the point s.

Notation I.129. Let α ∈ D(Rm) and I :=∏mi=1 Ii ∈ J (α).

(i) We define the time points

t0(α, θ) := 0, tn+1(α, θ) := inft > tn(α, θ),∥∥∆αt

∥∥2> θ, n ∈ N.

If t > tn(α, θ),∥∥∆αt

∥∥2> θ = ∅, then we set tn+1(α, θ) :=∞.

(ii) To the set I we associate the set of indices

JI :=i ∈ 1, . . . ,m, Ii 6= R. (I.35)

(iii) For the pair (α, I) we define the time points

s0(α, I) := 0, sn+1(α, I) := infs > sn(α, I),∆αis ∈ Ii for every i ∈ JI, n ∈ N. (I.36)

If s > sn(α, I),∆αis ∈ Ii for every i ∈ JI = ∅, then we set sn+1(α, I) :=∞.

The value of sn(α, I) marks the n−th time at which the value of ∆α lies in the set I and it iswell-defined since JI 6= ∅ for I ∈ J (α).

Example I.130. Let (tk)k∈N ⊂ R+, (xk)k∈N ⊂ R be such that tk −−→ t∞ and xk ↓ x∞ ∈ R \ 0.

Define γk· := xk1[tk,∞)(·), for every k ∈ N. By Lemma I.115.(iii), we have γkJ1(R)−−−−→ γ∞. On the other

hand, for I := (12x∞, 3

2x∞) holds s1(γk, I) = tk for all but finitely many k and s1(γ∞, I) = t∞, i.e.

s1(γk, I) −−−−→k→∞

s1(γ∞, I). Moreover, for w > x∞ we also have

∆γktk1(x∞,w)(∆γktk) = xk1(x∞,w)(x

k) = xk for all but finitely many k ∈ N,and

∆γ∞t∞1(x∞,w)(∆γ∞t∞) = x∞1(x∞,w)(x

∞) = 0.

Therefore, for R 3 x g7−→ x1(x∞,w)(x)

g(∆γks1(γk,I)) = ∆γktk1(x∞,w)(∆γktk) −−−−→

n→∞x∞ 6= 0 = ∆γ∞t∞1(x∞,w)(∆γ

∞t∞) = g(∆γ∞s1(α∞)),

and for this reason we cannot obtain the convergence

∆γktk1(x∞,w)(∆γntk)1(x∞,w)(·)

J1(R)−−−−→ ∆γ∞t∞1(x∞,w)(∆γ∞t∞)1(x∞,w)(·).

We can remedy the problem appearing in the previous example by not allowing elements of W (γ∞) tobe endpoints of the interval appearing in the indicator. Recall that W (γ) is at most countable, thereforewe are allowed to choose values for the endpoints from a dense subset of R \ 0. More precisely, we willchoose intervals from I(γ) so that only finitely many jumps occur on any every compact time-interval.

For the rest of this subsection we adopt a convenient abuse of notation and we will assume theextended positive real line [0,∞] endowed with a metric δ|·| which extends the usual metric of R+ and

for which δ|·|(∞,∞) = 0. Moreover, the convergence of a sequence (tk)k∈N ⊂ [0,∞] to the symbol ∞will be understood as follows: either tk = ∞ for every k ∈ N or for the subsequence (tkl)l∈N, where(kl)l∈N := k ∈ N, tkl < ∞, tkl −−→ ∞ in the usual sense. In this case, we will denote the convergenceof a sequence (tk)k∈N ⊂ [0,∞] to the symbol ∞ as usually by tk −−→∞.

Proposition I.131. Fix q, n ∈ N.(i) The function D(Rq) 3 α 7−→ tn(α, θ) ∈ R+ is continuous at each point α such that θ /∈ U(α).(ii) If tn(α, θ) < ∞, then the function D(Rq) 3 α 7−→ ∆αtn(α,θ) ∈ Rq is continuous at each point α

such that θ /∈ U(α).


Proof. See [41, Proposition VI.2.7].

The following proposition refines the above result.

Proposition I.132. Fix q, n ∈ N.(i) The function D(Rq) 3 α 7−→ sn(α, I) ∈ R+ is continuous at each point (α, I) ∈ D(Rq)× J (α).(ii) If sn(α, I) <∞, I ∈ J (α), then the function D(Rq) 3 α 7−→ ∆αsn(α,I) ∈ Rq is continuous.

Proof. Let (αk)k∈N be such that αkJ1(Rq)−−−−−→ α∞ and I :=

∏qi=1 Ii ∈ J (α∞). Observe that JI 6= ∅,

by definition of J (α∞). We define sk,n := sn(αk, I), for k ∈ N and n ∈ N.

(i) The convergence sk,0 −−−−→k→∞

s∞,0 holds by definition. Assume that the convergence sk,n −−−−→k→∞

s∞,n holds for some arbitrary n ∈ N. We will prove that the convergence sk,n+1 −−−−→k→∞

s∞,n+1 holds also.

For the following, fix a positive number θ∞I such that θ∞I /∈ ∪i∈JI|u|, u ∈W (α∞,i) and

θ∞I <1

qmin∪i∈JI|v|, v ∈ ∂Ii.22

Recalling the Notation I.129.(i), we have for every i ∈ JI thatt ∈ R+, |∆α∞,i| > θ∞I

=tl(α∞,i, θ∞I

)<∞, l ∈ N

. (I.37)

In other words, the settl(α∞,i, θ∞I

)< ∞, l ∈ N

is another way to write the set of times that α∞,i

exhibits a jump of height greater than θ∞I .

Case 1: s∞,n+1 <∞.By property (I.37), there exist unique li,n, li,n+1 ∈ N with li,n < li,n+1 such that

tli,n

(α∞,i, θ∞I ) = s∞,n and tli,n+1

(α∞,i, θ∞I ) = s∞,n+1 for every i ∈ JI . (I.38)

By Proposition I.131 and the above identities we obtain

tli,n

(αk,i, θ∞I ) −−−−→k→∞

s∞,n and ∆αk,itli,n

(αk,i,θ∞I )−−−−→k→∞

∆α∞,is∞,n for every i ∈ JI , (I.39)

as well as

tli,n+1

(αk,i, θ∞I ) −−−−→k→∞

s∞,n+1 and ∆αk,itli,n+1

(αk,i,θ∞I )−−−−→k→∞

∆α∞,is∞,n+1 for every i ∈ JI . (I.40)

Recall, now, the induction hypothesis, i.e. the convergence sk,n −−−−→k→∞

s∞,n is true. The Proposi-

tion I.121.(ii) guarantees that every sequence (rk)k∈N which converges to the time point s∞,n and it is

such that ∆αk,irk−−−−→k→∞

∆α∞,is∞,n , for some i = 1, . . . , q, then it finally coincides with (sk,n)n∈N. Therefore,

in view of the Convergence (I.39) we can obtain that(tli,n

(αk,i, θ∞I )− sk,n)k∈N ∈ c00(N), for every i ∈ JI , (I.41)

where c00(N) := (xm)m∈N ⊂ RN,∃m0 ∈ N such that xm = 0 for every m ≥ m0. Define for i ∈ JI

kn,i0 := maxk ∈ N, tli,n

(αk,i, θ∞I ) 6= sk,n <∞.Since JI is a finite set, the number

kn0 := maxkn,i0 , i ∈ JI

is well–defined and finite, therefore

sli,n

(αk,i, U) = sk,n, for every k > kn0 and for every i ∈ JI . (I.42)

Now, in view of Convergence (I.40) we can conclude the induction step once we prove the analogous to(I.41) for n+ 1 in place of n, i.e.

(tli,n+1

(αk,i, θ∞I )− sk,n+1)k∈N ∈ c00(N), for every i ∈ JI . (I.43)

At this point we distinguish two cases:

Case 1.1: For every i ∈ JI , holds li,n+1 = 1 + li,n.

22Recall that ∂A denotes the | · |−boundary of the set A ⊂ R.


By Proposition I.121.(ii), Convergence (I.40) and the convergence αkJ1(Rq)−−−−−→ α∞, we can conclude

that (tli,n+1

(αk,i, θ∞I )− tlj,n+1

(αk,j , θ∞I ))∈ c00(N), for every i, j ∈ JI . (I.44)

Therefore, we can fix hereinafter an index from JI and we will do so for µ := min JI , i.e. µ is theminimum element of JI . Define

kn+1,i0 := maxk ∈ N, tl

i,n+1

(αk,i, θ∞I ) 6= tlµ,n+1

(αk,µ, θ∞I ) for i ∈ JI \ µ.

By property (I.44), we obtain that kn+1,i0 < ∞ for every i ∈ JI \ µ. Since JI \ µ is a finite set, the

number

kn+10 := max

kn+1,i

0 , i ∈ JI \ µ

is well–defined and finite. Observe that

t1+li,n(αk,i, θ∞I ) = tli,n+1

(αk,i, θ∞I ) = tlµ,n+1

(αk,µ, θ∞I ), for k > kn+10 and for i ∈ JI , (I.45)

where the first equality holds by the assumption of this sub-case. Moreover, by Convergence (I.40) weobtain that

1 =∏i∈JI

1Ii(∆αk,i

t1+li,n (αk,i,θ∞I )), for all but finitely many k, (I.46)

since ∆α∞,is∞,n+1 lies in the interior of the open interval Ii, for every i ∈ JI .For notational convenience, we will assume that the above convergence holds for k > kn+1

0 . Therefore,for k > kn0 ∨ kn+1

0

sk,n+1 = infs > sk,n,∆αk,is ∈ Ii for every i ∈ JI = inf⋂i∈JI

s > sk,n,∆αk,is ∈ Ii

(I.41)

=k>kn0

inf⋂i∈JI

s > tl

i,n

(αk,i, θ∞I ),∆αk,is ∈ Ii

= inf⋂i∈JI

s ∈

(tli,n

(αk,i, θ∞I ), tlµ,n+1

(αk,i, θ∞I )],∆αk,is ∈ Ii

∧ inf

⋂i∈JI

s > tl

µ,n+1

(αk,i, θ∞I ),∆αk,is ∈ Ii

(I.45)=

k>kn+10

inf⋂i∈JI

s ∈

(tli,n

(αk,i, θ∞I ), t1+li,n(αk,i, θ∞I )],∆αk,is ∈ Ii

∧ inf

⋂i∈JI

s > t1+li,n(αk,i, θ∞I ),∆αk,is ∈ Ii

=

⋂i∈JI

t1+li,n(αk,i, θ∞I )

∧ inf

⋂i∈JI

s > t1+li,n(αk,i, θ∞I ),∆αk,is ∈ Ii

)=

⋂i∈JI

t1+li,n(αk,i, θ∞I )

(I.45)= tl

µ,n+1

(αk,i, θ∞I ) = tli,n+1

(αk,i, θ∞I ),

i.e. Property (I.43) indeed holds.

Case 1.2: There exists i ∈ JI for which holds li,n+1 > 1 + li,n.

Define JI : i ∈ JI , li,n+1 > 1 + li,n and fix ξi ∈ (li,n, li,n+1) ∩ N, for every i ∈ JI . Recall that byProposition I.131 holds

limk→∞

tξi

(αk,i, θ∞I ) = tξi

(α∞,i, θ∞I ) and limk→∞

∆αk,itξi (αk,i,θ∞I )

= ∆α∞,itξi (α∞,i,θ∞I )

. (I.47)

We can conclude that property (I.43) holds, if for every s ∈ (s∞,n, s∞,n+1) such that

limk→∞

tξi

(αk,i, θ∞I ) = s for every i ∈ JI (I.48)

holds

0 =∏

i∈JI\JI

1Ii(∆αk,is

) ∏i∈JI

1Ii(∆αk,i

tξi (αk,i,θ∞I )

)for all but finitely many k. (I.49)


assume to the contrary that

1 =∏

i∈JI\JI

1Ii(∆αk,is

) ∏i∈JI

1Ii(∆αk,i

tξi (αk,i,θ∞I )

)for all but finitely many k,

then in view of definition of s∞,n+1, we would also have s∞,n+1 = tξi

(α∞,i, θ∞I ) for every i ∈ JI . But,this contradicts property (I.38). The contradiction arises in view of

tξi

(α∞,i, θ∞I ) < tli,n+1

(αk,i, θ∞I ), since ξi < li,n+1.

Case 2: s∞,n+1 =∞.We distinguish two cases:

Case 2.1: s∞,n <∞Using the same arguments as the ones used in Property (I.38) for s∞,n, we can associate to tn(α∞,i, θI)

a unique natural number li,n such that Convergence (I.39) holds. Moreover, by definition of s∞,n+1 weobtain that

s > s∞,n,∆α∞,is ∈ Ii for every i ∈ JI = ∅,or, equivalently,

for every s > s∞,n there exists an i ∈ JI such that ∆α∞,i 6∈ Ii. (I.50)

Assume, now, that

lim infk→∞

sk,n+1 = s, for some s ∈ (sk,n,∞).

Therefore, there exists (kl)l∈N such that skl,n+1 −−−→l→∞

s. Equivalently, ∆αkl,iskl,n+1 ∈ Ii for all but finitely

many l for every i ∈ JI . By convergence αkJ1(Rq)−−−−−→ α∞, Proposition I.120.(iii) and convergence

skl,n+1 −−−→l→∞

s, we conclude that ∆α∞,is ∈ Ii, for every i ∈ JI . But this contradicts the assumption

s∞,n+1 =∞ in view of its equivalent form (I.50).

Case 2.2: s∞,n =∞.By definition of sk,n+1 holds

sk,n < sk,n+1 whenever sk,n <∞, for k ∈ N, n ∈ N,and

sk,n+1 =∞ whenever sk,n =∞, for k ∈ N, n ∈ N.

Therefore, induction hypothesis sk,n −−−−→k→∞

s∞,n and the above yield sk,n+1 −−−−→k→∞

∞ = s∞,n+1.

(ii) Assume that there exist n ∈ N and I ∈ J (α) such that s∞,n < ∞. By (i), we have that

sk,n −−−−→k→∞

s∞,n, which in conjunction with convergence αkJ1(Rq)−−−−−→ α∞ and Proposition I.120 implies

that

∆αksk,n −−−−→k→∞

∆α∞s∞,n .

Corollary I.133. Let αkJ1(Rq)−−−−−→ α∞ and I ∈ J (α∞). Define

n :=

maxn ∈ N, sn(α∞, I) <∞, if n ∈ N, sn(α∞, I) <∞ is non–empty and finite,

∞, otherwise.

Then for every function g : Rq → R which is continuous on

C :=

q∏i=1

Ai, where Ai :=

W (α∞,i), if i ∈ JI ,W (α∞,i) ∪ 0, if i ∈ 1, . . . , q \ JI ,

and for every 0 ≤ n ≤ n holds

g(∆αksn(αk,I)) −−−−→k→∞

g(∆α∞sn(α∞,I)).


In particular, for the cadlag functions

βk· := g(∆αksn(αk,I))1[sn(αk,I),∞)(·), for k ∈ N,

the convergence βkJ1(R)−−−−→ β∞ holds.

Proof. Fix an n ∈ N such that n ≤ n. By Proposition I.132.(ii) holds

∆αksn(αk,I) −−−−→k→∞

∆α∞sn(α,∞,I), where ∆α∞,isn(α∞,I) ∈

W (α∞,i), if i ∈ JI ,W (α∞,i) ∪ 0, if i ∈ 1, . . . , q \ JI .

Therefore, by definition of the time sn(α∞, I) and of the set C holds

g(∆αksn(αk,I)) −−−−→k→∞

g(∆α∞sn(α∞,I)). (I.51)

By Proposition I.132.(i), the above convergence and Lemma I.115 we obtain the convergence

βkJ1(R)−−−−→ β∞.

Proposition I.134. Fix some subset I :=∏qi=1 Ii of Rq and a function g : Rq → R. Define the map

D(Rq) 3 α 7−→ α[g, I] := (α1, . . . , αq, αg,I) ∈ D(Rq+1),

where

αg,I· :=∑

0<t≤·

g(∆αt)1I(∆αt). (I.52)

Then, the map α[g, I] is J1−continuous at each point α for which I ∈ J (α) and for each function g whichis continuous on the set

C :=

q∏i=1

Ai, where Ai :=

W (αi), if i ∈ JI ,W (αi) ∪ 0, if i ∈ 1, . . . , q \ JI .

(I.53)

Proof. The arguments are similar to those in the proof of [41, Corollary VI.2.8], but for the conve-nience of the reader we will present the proof below.

Let (αk)k∈N ⊂ D(Rq) be such that αkJ1(Rq)−−−−−→ α∞, I :=

∏qi=1 Ii ∈ J (α∞) and a function g : Rq → R

which is continuous on the set

C :=

q∏i=1

Ai, where Ai :=

W (α∞,i), if i ∈ JI ,W (α∞,i) ∪ 0, if i ∈ 1, . . . , q \ JI .

For notational convenience, denote sk,p := sp(αk, I), for k ∈ N and p ∈ N, and Rq 3 xgI7−→ g(x)1I(x).

Recall that for every i ∈ JI holds ∂Ii ∩W (α∞,i) = ∅, therefore gI

remains continuous on C.

Step 1: Let

p :=

0, if p ∈ N, s∞,p <∞ = ∅,maxp ∈ N, s∞,p <∞, if p ∈ N, s∞,p <∞ is non–empty and finite,

∞, if p ∈ N, s∞,p <∞ is non–empty and infinite,

• If p = 0, then by convergence αkJ1(R)−−−−→ α∞ holds in particular αk0 −−−−→

k→∞α∞0 . Since s0(αk, I) =

s0(α∞, I) = 0 by definition, we can conclude.• If p <∞, using Corollary I.133 we get that for every p ∈ N with p ≤ p it holds that

gI(∆αksk,p)1[sk,p,∞)(·)

J1(R)−−−−→k→∞

gI(∆α∞s∞,p)1[s∞,p,∞)(·).

By Proposition I.132 we obtain that for p > p it holds sk,p −−−−→k→∞

∞. Therefore, by Lemma I.115.(i)

we conclude that for every p > p


J1(R)−−−−→k→∞

0. (I.54)


Using now the fact that s∞,p−1 < s∞,p, for every 1 ≤ p ≤ p, in conjunction with Proposition I.120 wecan prove by induction that, for every 1 ≤ q ≤ p, the following holds∑

1≤p≤q


J1(R)−−−−→k→∞

∑1≤p≤q

gI(∆α∞s∞,p)1[s∞,p,∞)(·).

Analogously, we can prove by induction that for q > p holds∑p<p≤q


J1(R)−−−−→k→∞

0,

by the continuity of the limit in Convergence (I.54) for every p > p. In view of Proposition I.120 we cancombine the two last convergences, to obtain the convergence of their sum. In particular, we can havefor every q ∈ N ∑

1≤p≤q


J1(R)−−−−→k→∞

∑1≤p≤q

gI(∆α∞s∞,p)1[s∞,p,∞)(·). (I.55)

• If p =∞, we can use the arguments of the first part of the previous case (i.e. when p ≤ p) to proveby induction that for every q ≥ 1 holds∑

1≤p≤q


J1(R)−−−−→k→∞

∑1≤p≤q

gI(∆α∞s∞,p)1[s∞,p,∞)(·).

Step 2: Let now N > 0 and define pN

:= minp ∈ N, s∞,p > N. By the definition of (s∞,p)p∈Nand the fact that α∞ ∈ D(Rq) we can easily conclude that s∞,p −−−→

p→∞∞, hence p

Nis well–defined.

Then, by Proposition I.132 the convergence sk,pN −−−−→k→∞

s∞,pN holds. Observe that we do not need to

assume s∞,pN < ∞ at this point. Using the last convergence, there exists k0 ∈ N such that for everyk ≥ k0 holds sk,pN > N.

Define αk,g,I := (αk)g,I

for k ∈ N, (see again (I.52)) and observe now that

α∞,g,I· 1[0,s∞,pN )(·) =∑

0<t≤·

gI(∆α∞t )1[0,s∞,pN )(·) =

∑0<t<s∞,pN ∧·

gI(∆α∞t )

=∑

1≤p≤pN−1

gI(∆α∞s∞,p)1[s∞,p,∞)(·).

Moreover, we have for k ≥ k0

αk,g,I· 1[0,sk,pN )(·) =∑

0<t≤·

gI(∆αkt )1[0,sk,pN )(·) =

∑0≤t<sk,pN ∧·

gI(∆αkt )

=∑

1≤p≤pN−1

gI(∆αksk,p)1[sk,p,∞)(·).

By Convergence (I.55) and for the chosen pN

there exists a sequence (λN,k)k∈N ⊂ Λ, for Λ as defined inDefinition I.108.(ii), for which the sequence

(∑1≤p≤pN g

I(∆αksk,p)1[sk,p,∞)(·)

)k∈N satisfies Theorem I.109.

In view of the last equalities on [0, N ] and for k ≥ k0 we have that

sups∈[0,N ]

|αk,g,IλN,k(s)

− α∞,g,Is | ∨ sups∈[0,N ]

|λN,k(s)− s|

= sups∈[0,N ]

∣∣∣ ∑1≤p≤p

N−1

gI(∆αksk,p)1[sk,p,∞)(λ

N,k(s))−∑

1≤p≤pN−1

gI(∆α∞s∞,p)1[s∞,p,∞)(s)

∣∣∣∨ sups∈[0,N ]

|λN,k(s)− s| n→∞−−−−−→(I.55)

0,

Now that we have the family of sequences (λN,k)k∈N, N ∈ N, we can follow the same arguments as inthe proof of [41, Lemma VI.1.31] in order to construct a sequence (λk)k∈N ⊂ Λ, for which the sequence(αk,g,I)k∈N satisfies Theorem I.109. Therefore we can conclude that

αk,g,IJ1(R)−−−−−→k→∞

α∞,g,I .


Now we have only to observe that α∞,g,I may have a jump at t only if t = sp(α∞, I), for some p ∈ N.Hence, by combining Proposition I.120 and Proposition I.132 we can conclude the joint convergence

(αk,1, . . . , αk,q, αk,g,I)J1(Rq+1)−−−−−−−−→k→∞

(α∞,1, . . . , α∞,q, α∞,g,I). (I.56)

I.6.2. Application to cadlag processes. In this section we collect some results regarding con-vergence of random variables taking their values in the Polish space D(E). We will follow closely Jacodand Shiryaev [41, Section VI.3]. For this section, we additionally endow the arbitrary probability space(Ω,G,P) with an arbitrary sequence of filtrations (Gk)k∈N.

Definition I.135. Let (Mk)k∈N be an arbitrary sequence such that Mk is an E−valued cadlag process,

for every k ∈ N.

(i) The sequence (Mk)k∈N converges in probability under the J1(E)−topology to M∞ if

P(δJ1(E)(M

k,M∞) > ε)−−−−−−→k→∞

0, for every ε > 0,

and we denote it by Mk (J1(E),P)−−−−−−−→M∞.(ii) Let ϑ ∈ 1, 2. The sequence (Mk)k∈N converges in Lϑ−mean under the J1(E)−topology to M∞

if

E[(δJ1(E)(M

k,M∞))ϑ] −−−−−−→

k→∞0,

and we denote it by Mk (J1(E),Lϑ)−−−−−−−−→M∞.

(iii) Analogously, we denote byMk (lu,P)−−−−→M∞, resp. Mk (lu,Lϑ)−−−−−→M∞, the convergence in probability,resp. in Lϑ−mean, under the locally uniform topology.

(iv) Moreover, let (Nk,1)k∈N be a sequence of E1−valued and cadlag processes and (Nk,2)k∈N be asequence of E2−valued and cadlag processes. For ϑ1, ϑ2 ∈ 1, 2 we will write

(Nk,1, Nk,2)(J1(E1×E2),Lϑ1×Lϑ2 )−−−−−−−−−−−−−−−−−→ (N∞,1, N∞,1),

if the following convergence hold

(Nk,1, Nk,2)(J1(E1×E2),P))−−−−−−−−−−−−→ (M∞, N∞), Nk,1 (J1(E1),Lϑ1 )−−−−−−−−−→ N∞,1 and Nk,2 (J1(E2),Lϑ2 )−−−−−−−−−→ N∞,2.

Definition I.136. Let (Γ, δΓ) be a metric space.

(i) The set of probability measures defined on(Γ,B(Γ)

)is denoted by P(Γ).

(ii) Let A ⊂P(Γ). The set A will be called tight if for every ε > 0 there exists a compact set K inΓ such that Q(Γ \K) ≤ ε for every Q ∈ A .

(iii) The law of a Γ-valued random variable Ξ, which will be denoted by L(Ξ), is the probability measureon(Γ,B(Γ)

)defined for every A ∈ B(Γ) by L(Ξ)(A) := P

(ω ∈ Ω,Ξ(ω) ∈ A).

(iv) Let (Ξk)k∈N be a sequence of Γ−valued random variables. We will say that Ξk converges weakly

to Ξ∞ if for every f :(Γ, δΓ

)→ (R, | · |) continuous and bounded holds

E[f(Ξk)

]−−→ E

[f(Ξ∞)

].

We will denote the above convergence by ΞkL−−→ Ξ∞ or L(Ξk)

w−−→ L(Ξ∞) interchangeably. The randomvariable Ξ∞ will be called the weak limit (point) of the sequence (Ξk)k∈N.

(v) Let (Ξk)k∈N be a sequence of Γ−valued random variables. The sequence (Ξk)k∈N will be called

tight if the sequence of the associated laws(L(Xk)

)k∈N is tight.

Remark I.137. The reader may recall the Dunford–Pettis Compactness Criterion, see Theorem I.26,which deals with relatively compact subsets of P(R).

Lemma I.138. Let X be a real-valued and cadlag process. The set

V (X) :=u ∈ R \ 0,P

([∆Xt = u, for some t > 0]

)> 0

is at most countable.

Proof. It is an immediate consequence of [41, Lemma VI.3.12].

We state the well-known Prokhorov’s theorem in the case of Polish spaces.


Theorem I.139 (Prokhorov). Let Γ be a separable and complete metric space. The set A ⊂ P(Γ) isrelatively compact if and only if A is tight.

Proof. See Prokhorov [57, Theorem 1.12].

Proposition I.140. Let (Xk)k∈N be a sequence of Rq−valued cadlag processes such that Xk L−−→ X∞

in D(Rq). Then, (Xkt1 , . . . , X

ktn)

L−−→ (X∞t1 , . . . , X∞tn ) in Rqn, for every ti ∈

t ∈ R+,P([∆X∞t 6= 0]) > 0

and for every n ∈ N.


Theorem I.141. Let (Xk)k∈N be a sequence of E−valued cadlag processes. Then, the following areequivalent:

(i) Xk (J1(E),P)−−−−−−−→ X∞.(ii) For every subsequence (Xkl)l∈N there exists a further subsequence (Xklm )m∈N such that

XklmJ1(E)−−−−−→ X∞ P− a.s..

Proof. See Dudley [28, Theorem 9.2.1], which can be applied in this case because the Skorokhodspace is Polish under the J1−topology.

Definition I.142. A sequence (Xk)k∈N of processes is called C−tight if it is tight and if all weak limitpoints of the sequence

(L(Xk)

)k∈N are laws of continuous processes.

Lemma I.143. Let (Xk)k∈N be an Rp−valued C−tight sequence and (Y k)k∈N be an Rq−valued tight,resp. C−tight, sequence. Then:

(i) If p = q, then (Xk + Y k)k∈N is tight, resp. C−tight.(ii) (Xk, Y k)k∈N is an Rp+q−valued tight, resp. C−tight, sequence.

Proof. See Jacod and Shiryaev [41, Corollary VI.3.33].

Definition I.144. Let A and B two increasing processes. We say that B strongly majorizes A if theprocess B −A is itself increasing.

In the following proposition the probability space is endowed with a sequence of filtrations (Gk)k∈N.

Proposition I.145. Let (Ak)k∈N, (Bk)k∈N be sequences of cadlag processes. Assume, moreover, thatAk, Bk are increasing and such that Bk strongly majorizes Ak for every k ∈ N. If (Bk)k∈N is tight, resp.C−tight, then A is also tight, resp. C−tight.


The following theorem provides a sufficient condition for a sequence of Rq−valued square-integrableGk−martingales to be tight.

Theorem I.146. Let (Xk)k∈N be a sequence such that Xk −Xk0 ∈ H2(Gk;Rq) for every k ∈ N. If

(i) the sequence (Xk0 )k∈N is tight in R and

(ii) the sequence(Tr[〈Xk〉

])k∈N is C−tight in D(R),

then the sequence (Xk)k∈N is tight in D(Rq).

Proof. See Jacod and Shiryaev [41, Theorem VI.4.13].

The last mentioned theorem is proven in Rebolledo [59] and it is a consequence of Aldous’ Criterionfor Tightness,; for more details for the aforementioned criterion the reader may consult Aldous [2] and[41, Section VI.4a]. The role of Theorem I.146 in Chapter III will not be as evident as its importancemay deserve, since it will be applied in a single point of the proof of a lemma among many lemmata; seeLemma III.31. For this reason, we need to specifically comment at this point, that it is an extremelyconvenient result, which enabled us to obtain the tightness of a sequence of joint laws.

We proceed, now, to another useful result. The message of the next theorem can be loosely describedas follows: the weak limit of a sequence of martingales will be a uniformly integrable martingale withrespect to its natural filtration as soon as the family of random variables obtained by the convergentsequence of martingales is uniformly integrable. It is well-known that the martingale property can beweakened to the submartingale property.


Proposition I.147. Let (Hk)k∈N be a sequence of E1−valued cadlag processes, where Hk is Gk−adapted

for every k ∈ N.23 Assume, moreover, that (Θk)k∈N is a sequence such that Θk is an E2−valued cadlag

Gk−martingale for k ∈ N.24 Furthermore, let Θ∞ be a cadlag adapted process defined on the canonicalspace

(D(Rq),D(Rq),D(Rq)

)and let D ⊂ R+ be dense. Assume that:

(i) The family‖Θk

t ‖1, t ∈ R+ and k ∈ N


(ii) Hk L−−→ H∞.(iii) For all t ∈ D, D(Rq) 3 α 7−→ Θ∞t (α) is L(H∞)− a.s. continuous and

(iv) Θkt −Θ∞t Hk (|·|,P)−−−−−→ 0 for all t ∈ D.

Then the process Θ∞ H∞ is a martingale with respect to the filtration generated by H∞.

Proof. See Jacod and Shiryaev [41, Proposition IX.1.12]. The uniform integrability of the limitmartingale is a consequence of Dunford–Pettis Compactness Criterion, see Theorem I.26.

We will close the section with the introduction of another important notion for limit theorems andthe presentation of some classical related results.

Notation I.148. For every k ∈ N, H k will denote the set of Gk−predictable elementary processesbounded by 1. In other words,

H k :=H : (Ω× R+,PGk) −−→

(R+,B(R+)

), Ht = h010(t) +

p∑i=1

hi1(si,si+1](t) with

p ∈ N, 0 = s0 < s1 < . . . , < sp+1 and hi is Gksi −measurable with |hi| ≤ 1

Definition I.149. A sequence (Xk)k∈N of Rq−valued cadlag processes, where Xk is Gk−adapted foreach k ∈ N, is said to be predictably uniformly tight , in short P-UT, if for every t > 0 the family ofrandom variables

∑qi=1(Hk,i ·Xk,i)t, k ∈ N and Hk,i ∈H k

is tight in R,25 that is

limK↑∞

sup

P(∣∣∣∣ q∑

i=1

(Hk,i ·Xk,i)t

∣∣∣∣ > K

), Hk,i ∈H k and k ∈ N

= 0.

Remark I.150. It is immediate by the definition that, if the sequences (Xk)k∈N, (Y k)k∈N are P-UTthen so is (Xk + Y k)k∈N.

Proposition I.151. Let (V k)k∈N be such that V k ∈ V(Gk;Rp) for every k ∈ N.

(i) If(Var(V k)t

)k∈N is tight in R for every t ∈ R+, then (V k)k∈N is P-UT.

(ii) If, moreover, V k ∈ Vpred(Gk;Rp) for every k ∈ N and the sequence(Var(V k)t

)k∈N is P-UT, then(

Var(V k)t)k∈N is tight in R for every t ∈ R+.

Proof. (i) It is immediate by the definition of the variation process and the fact that the elementsof H k are bounded by 1, for every k ∈ N.

(ii) This is Jacod and Shiryaev [41, Proposition VI.6.12]

Theorem I.152. Let the sequence (Sk)k∈N of Rq−valued cadlag processes and assume it is P-UT.

If SkL−−→ S∞, then (Sk, [Sk])

L−−→ (S∞, [S∞]) in D(Rq × Rq×q). If the former convergence holds inprobability, then the later holds in probability as well.

Proof. See Jacod and Shiryaev [41, Theorem VI.6.26].

Proposition I.153. Let the sequence (Xk)k∈N of Rq−valued processes be such that

(i) Xk is Gk−martingale, for every k ∈ N,(ii) supk∈N E

[sup0≤s≤t ‖Xk

s ‖22]<∞, for every t ∈ R+ and

(iii) Xk L−−→ X∞.

23We do not require the property k =∞.24As in 23, we do not require the property for k =∞.25In the literature, it is also used the term bounded in probability, with the obvious interpretation.


Then (Xk)k∈N is P-UT and thus (Xk, [Xk])L−−→ (X∞, [X∞]). If the convergence in (iii) holds in probabil-

ity, then the above convergence is also true in probability, i.e. (Xk, [Xk])(J1(Rq×Rq×q),P)−−−−−−−−−−−−→ (X∞, [X∞]).

Proof. We will apply Jacod and Shiryaev [41, Corollary VI.6.30]. We need only to prove thatsupk∈N E

[sups≤t ‖∆Xk

s ‖2]< ∞, for every t ∈ R+. To this end let us fix a t ∈ R+. We observe that in

view of the inequality

supk∈N

∥∥∥ sups≤t‖∆Xk

s ‖2∥∥∥L1(G;R)

≤ supk∈N

∥∥∥ sups≤t‖∆Xk

s ‖2∥∥∥L2(G;R)

= supk∈N

E[

sups≤t‖∆Xk

s ‖22] 1

2

it is sufficient to prove that the right-hand side is finite. For the equality we have used that the functionR+ 3 x 7→ x2 ∈ R+ is continuous. Now, we have

supk∈N

E[sups≤t‖∆Xk

s ‖22]

= supk∈N

E

[sup

0≤s≤t

q∑i=1

|∆Xk,is |2

]≤ sup

k∈NE

[sup

0≤s≤t

(2

q∑i=1

|Xk,is |2 + 2

q∑i=1

|Xk,is− |2

)]

≤ supk∈N

2

q∑i=1

E[(

sup0≤s≤t

|Xk,is |)2]

+ 2

q∑i=1

E[(

sup0≤s≤t

|Xk,is− |)2] ≤ 4 sup

k∈N

q∑i=1

E[(

sup0≤s≤t

|Xk,is |)2]

= 4 supk∈N

E[ q∑i=1

sup0≤s≤t

|Xk,is |2

]≤ 4q sup

k∈NE[

sup0≤s≤t

‖Xk,is ‖22

] (ii)< ∞,

where in the third inequality we have used that the processes are cadlag. therefore, (Xk)k∈N is P-UT.For the convergence in probability, we apply [41, Theorem VI.6.22 (c)].

I.6.3. Weak convergence of filtrations. The purpose of this subsection is to introduce the notionfor convergence of σ−algabrae and of filtrations we are going to make use of in the sequel. In theliterature, such notions have been proposed by Aldous [3] and Hoover [38]. However, we are going to usethat of Coquet et al. [23], which weakens the notion introduced by Hoover. Let us, now, proceed to theintroduction of these notions.

Definition I.154. (i) A sequence of σ−algebrae (Gk)k∈N converges weakly to the σ−algebra G∞ if,for every ξ ∈ L1(G∞;R), we have

E[ξ|Gk]P−−→ E[ξ|G∞].

We denote the weak convergence of σ−algebrae by Gk w−−→ G∞.(ii) A sequence of filtrations

(Gk := (Gkt )t∈R+

)k∈N converges weakly to G∞ := (G∞t )t∈R+ , if, for every

ξ ∈ L1(G∞∞ ;R), we have

E[ξ|Gk· ](J1(R),P)−−−−−−−→ E[ξ|G∞· ].

We denote the weak convergence of the filtrations by Gk w−−→ G∞.(iii) Consider the sequence

((Mk,Gk)

)k∈N, where Mk is an E−valued cadlag process and Gk is a

filtration, for any k ∈ N. The sequence((Mk,Gk)

)k∈N converges in the extended sense to (M∞,G∞) if

for every ξ ∈ L1(G∞∞ ;R), (Mk,E[ξ|Gk· ]

) (J1(E×R),P)−−−−−−−−−→(M∞,E[ξ|G∞· ]

). (I.57)

We denote the convergence in the extended sense by(Mk,Gk

) ext−−−→ (M∞,G∞).

Remark I.155. For the definition of weak convergence of filtrations, we could have used only randomvariables ξ of the form 1A, for A ∈ G∞∞ . Indeed, the two definitions are equivalent; see Coquet et al. [23,Remark 1.1)].

The following result, which is Hoover [38, Theorem 7.4], provides a sufficient condition for weakconvergence of σ−algebrae which are generated by random variables.

Example I.156. Let (ξk)k∈N be a sequence of random variables such that ξkP−−→ ξ∞. Then the con-

vergence σ(ξk)w−−→ σ(ξ∞) holds, where σ(ψ) denotes the σ−algebra generated by the random variable

ψ.

In the next example, which is Coquet et al. [23, Proposition 2], a sufficient condition for the weakconvergence of the natural filtrations of stochastic processes is provided.


Example I.157. Let Mk be a process with independent increments, for every k ∈ N. If Mk (J1(Rq),P)−−−−−−−→M∞, then GMk w−−→ GM∞ .

In the remainder of this section, we fix an arbitrary sequence of filtrations (Gk)k∈N on (Ω,G,P), with

Gk := (Gkt )t∈R+, and an arbitrary sequence (Mk)k∈N, where Mk ∈ M(Gk;Rq) for every k ∈ N. Recall

that the random variables Mk∞ := limt→∞Mk

t are well–defined P− a.s., and Mk∞ ∈ L1(Ω,Gk∞,P;Rq), for

k ∈ N; see Theorem I.57.Next, we would like to discuss how to deduce the extended convergence of martingales and filtrations

from individual convergence results. Such properties have already been obtained by Memin [53, Propo-sition 1.(iii)], where he refers to Coquet et al. [23, Proposition 7] for the proof. However, the authors in[23] proved the result under the additional assumption that the processes are adapted to their naturalfiltrations. Moreover, they consider a finite time horizon T , which gives the time point T a special rolefor the J1(R)−topology on D([0, T ];R); see also Jacod and Shiryaev [41, Remark VI.1.10]. In addition,

in Coquet et al. [23, Remark 1.2)] the convergence Mk∞

L1(G∞∞ ;Rq)−−−−−−−−→ M∞∞ is assumed, although it is notnecessary (note that we have translated their results into our notation). This is restrictive, in the sensethat they have to assume in addition the G∞∞−measurability of Mk

∞, for each k ∈ N.

We present below, for the sake of completeness, the statement and proof of the aforementioned resuts

for the infinite time horizon case, under the condition Mk∞

L1(G;Rq)−−−−−−−→M∞∞ .

Proposition I.158. Assume the convergence Mk∞

L1(Ω,G,P;Rq)−−−−−−−−−→ M∞∞ holds. Then, the convergence

Gk w−−→ G∞ is equivalent to the convergence (Mk,Gk)ext−−−→ (M∞,G∞).

Proof. The first step is to show that the L1 convergence of (Mk∞)k together with the weak conver-

gence of the filtrations implies the convergence of the martingales in the J1(Rq)−topology. Let ε > 0 and

Gk w−−→ G∞, then

P(dJ1(Rq)(M

k,M∞) > ε)

≤ P(dJ1(Rq)

(Mk,E[M∞∞ |Gk· ]

)>ε

2

)+ P

(dJ1(Rq)

(E[M∞∞ |Gk· ],M∞

)>ε

2

)≤ P

(sup

t∈[0,∞)

∣∣Mkt − E[M∞∞ |Gk· ]

∣∣ > ε

2

)+ P

(dJ1(Rq)

(E[M∞∞ |Gk· ],M∞

)>ε

2

)≤ 2

εE[|Mk∞ −M∞∞ |

]+ P

(dJ1(Rq)

(E[M∞∞ |Gk· ],E[M∞∞ |G∞· ]

)>ε

2

)k→∞−−−−−→ 0, (I.58)

where the first summand converges to 0 by assumption and the second one by the weak convergence ofthe filtrations. Let us point out that for the second inequality we have used that for α, β ∈ D(Rq) holds

dJ1(Rq)(α, β) ≤ dlu(α, β) ≤ d‖·‖∞(α, β),

by the definition of the metrics, while for the third inequality we used Doob’s martingale inequality.

The next step is to apply Lemma I.124 to (Nk)k∈N := (Mk,E[ξ|Gk· ])k∈N, for ξ ∈ L1(G∞∞ ;R), in

order to obtain the convergence in the extended sense. The J1(R)−convergence of each (Nk,i)k∈N, for

i = 1, . . . , q, and of the partial sums (∑pi=1N

k,i)k∈N, for p = 1, . . . , q follows from the previous step and

Lemma I.124. Moreover, the J1(R)−convergence of (Nk,q+1)k∈N follows from the definition of the weak

convergence of filtrations. Hence, we just have to show the J1(R)−convergence of (∑q+1i=1 N

k,i)k∈N.

By assumption, we have∑qm=1M

k,m∞ + ξ

L1(G;R)−−−−−−→k→∞

∑qm=1M

∞,m∞ + ξ, and arguing as in (I.58) we

obtain

E[ q∑m=1

Mk,m∞ + ξ

∣∣∣Gk· ] (J1(R),P)−−−−−−−→k→∞

E[ q∑m=1

M∞,m∞ + ξ∣∣∣G∞· ].

Moreover, by the linearity of conditional expectations, we get that

q∑m=1

E[Mk,m∞ |Gk· ] + E[ξ|Gk· ]

(J1(R),P)−−−−−−−→k→∞

q∑m=1

E[Mm∞|G∞· ] + E[ξ|G∞· ].

The converse statement is trivial.


Now we are going to present the cornerstones for the convergence in the extended sense, which areMemin [53, Theorem 11, Corollary 12]. Here we state and prove them in the multidimensional case.

Theorem I.159. Let (Sk)k∈N be a sequence of Rq−valued Gk−special semimartingales with Gk−canonical

decomposition Sk = Sk0 +Mk +Ak, for every k ∈ N. Assume that S∞ is G∞−quasi–left–continuous andthe following properties hold

(i) The sequence([Sk,i]

1/2∞)k∈N is uniformly integrable, for every i = 1, . . . , q.

(ii) The sequence (‖Var(Ak)∞‖1)k∈N is tight.

(iii) The extended convergence (Sk,Gk)ext−−−→ (S∞,G∞) holds.

Then

(Sk,Mk, Ak)(J1(R3q),P)−−−−−−−−→ (S∞,M∞, A∞).

Proof. By Memin [53, Theorem 11], we obtain for every i = 1, . . . , q the following convergence

(Sk,i,Mk,i, Ak,i)(J1(R3),P)−−−−−−−−→ (S∞,i,M∞,i, A∞,i).

Then, by assumption Sk(J1(Rq),P)−−−−−−−→ S∞, and using Corollary I.125 and Remark I.123 we obtain the

required result.

Theorem I.160. Let Mk ∈ H2(Gk,∞;Rq) for any k ∈ N and M∞ be G∞−quasi–left–continuous. Ifthe following convergence hold

(Mk,Gk)ext−−−→ (M∞,G∞) and Mk

∞L2(G;Rq)−−−−−−−−→M∞∞ ,

then

(i) (Mk, [Mk], 〈Mk〉)(J1(Rq×Rq×q×Rq×q),P)−−−−−−−−−−−−−−−−−→ (M∞, [M∞], 〈M∞〉),

(ii) for every i = 1, . . . , q, we have

[Mk,i]∞L1(G;R)−−−−−−−→k→∞

[M∞,i]∞ and 〈Mk,i〉∞L1(G;R)−−−−−−−→k→∞

〈M∞,i〉∞.

Proof. (i) By Memin [53, Corollary 12], we obtain for every i = 1, . . . , q the convergence

(Mk,i, [Mk]ii)>(J1(R2),P)−−−−−−−−→ (M∞,i, [M∞]ii)>,

which in conjunction with Corollary I.125 and the convergence Mk (J1(Rq),P)−−−−−−−→M∞ implies(Mk, [Mk]11, . . . , [Mk]qq

) (J1(R2q),P)−−−−−−−−→

(M∞, [M∞]11, . . . , [M∞]qq

)(I.59)

On the other hand, let i, j ∈ 1, . . . , q with i 6= j. Observe, now, that the sequence (Mk)k∈N satisfiesthe requirements of Proposition I.153. Therefore, the sequence is P-UT and by Theorem I.152, we obtain

(Mk, [Mk])(J1(Rq×Rq×q),P)−−−−−−−−−−−−→ (M∞, [M∞]). (I.60)

Let us now show that the predictable quadratic variations converge as well. By Memin [53, Corollary12], we have for i = 1, . . . , q

〈Mk〉ii (J1(R),P)−−−−−−−→ 〈M∞〉ii. (I.61)

Moreover the convergence Mk∞

(J1(R2),L2)−−−−−−−−→ M∞∞ implies in particular, for every i, j = 1, 2, . . . , q with

i 6= j, that

Mk,i∞ +Mk,j

∞(J1(R2),L2)−−−−−−−−→M∞,i∞ +M∞,j∞ . (I.62)

In view of (I.59) and (I.62), we can apply Memin [53, Corollary 12] to (Mk,i + Mk,j)k∈N and (Mk,i −Mk,j)k∈N, for every i, j = 1, 2, . . . , q with i 6= j. Therefore we get that

〈Mk,i +Mk,j〉 (J1(R),P)−−−−−−−→ 〈M∞,i +M∞,j〉,

and 〈Mk,i −Mk,j〉 (J1(R),P)−−−−−−−→ 〈M∞,i −M∞,j〉.


Now recall that M∞ is quasi–left–continuous, which implies that the processes 〈M∞,i + M∞,j〉 and〈M∞,i −M∞,j〉 are continuous for every i, j = 1, 2, . . . , q with i 6= j. Therefore, by Theorem I.62 andProposition I.120 and the last results we obtain

〈Mk〉ij =1

4(〈Mk,i +Mk,j〉 − 〈Mk,i −Mk,j〉) (J1(R),P)−−−−−−−→

1

4(〈M∞,i +M∞,j〉 − 〈M∞,i −M∞,j〉) = 〈M∞〉ij .

(I.63)

Concluding, by (I.60), (I.61), (I.63) and due to the continuity of 〈M∞〉 we have

(Mk, [Mk], 〈Mk〉)(J1(Rq×Rq×q×Rq×q),P)−−−−−−−−−−−−−−−−−→ (M∞, [M∞], 〈M∞〉).

(ii) Let i = 1, . . . , q. The sequence (Mk,i)k∈N satisfies the conditions of Memin [53, Corollary 12]. Inthe middle of the proof of the aforementioned corollary, we can find the required convergence.

I.7. Martingale representation of square-integrable martingales

This section deals with the martingale representation of a square integrable martingale with respectto a pair of square-integrable martingales. In order to provide a better description of how we willunderstand a martingale representation let us provide the classical result [41, Lemma III.4.24]. Weare going to translate the general framework of the aforementioned lemma into the square-integrablecase. Now, given a pair X,Y where X ∈ H2(G;R`) and Y ∈ H2(G;R), there exist a unique, up toindistinguishability, decomposition of the martingale Y with respect to the natural pair of X whichsatisfies the following:

Y = Z ·Xc + U ? µ(Xd,G) +N, (I.64)

where Z ∈ H2(G, Xc;R`), U ∈ H2(G, µXd ;R) and

〈N c, Xc,i〉 = 0 for every i = 1, . . . , ` and MµXd[∆N

∣∣PG] = 0.

Observe that the above properties imply that N ⊥⊥ H ·Xc+W ?µ(Xd,G) for every H ∈ H2(G, Xc;R`) and

W ∈ H2(G, µXd ;R), a property which justifies the alternative term orthogonal decomposition of Y withrespect to X. Before we proceed, let us provide the following definition for the representation propertyof a martingale X with respect to a filtration.

Definition I.161. Let X ∈ H2(G;R). We will say that the martingale X possesses the G−predictablerepresentation property if for every real-valued G−martingale Y holds

Y c ∈ L2(G, Xc;R) and Y d ∈ K2(G, µXd

;R).

In other words, there exist Z ∈ H2(G, Xc;R1×`) and U ∈ H2(G, µXd ;R) such that

Y = Y0 + Z ·Xc + U ? µ(Xd,G).

Having in mind that we want to develop a framework suitable also for discrete-time approximations,the presence of a continuous martingale in the representation rules out this possibility. Consequently,we would like to be able to construct an analogous orthogonal decomposition of Y with respect to apair (X, X\) ∈ H2(G;R`) × H2,d(G;R`). Under this framework, the martingale X will play the roleof the integrator of the Ito stochastic integral and the purely discontinuous martingale X\ will providethe integer-valued random measure. Since X may also have a purely discontinuous part, it is naturalto ask for the orthogonality of the Ito integral and the stochastic integral with respect to the integer-valued random measure that will appear in the decomposition. After these comments, we can providethe following definition.

Definition I.162. Let (X, X\) ∈ H2(G;R`)×H2,d(G;R`)26 and Y ∈ H2(G;Rp). The decomposition

Y = Y0 + Z ·X + U ? µ(X\,G) +N

will be called the orthogonal decomposition of Y with respect to (G, X, µX\) or the representation of the

martingale Y with respect to (G, X, µX\) if

(i) Z ∈ H2(G, X;Rp×`) and U ∈ H2(G, µX\ ;Rp),(ii) 〈Z ·X, U ? µ(X\,G)〉 = 0 and

26Observe that the purely discontinuous part of X\ is X\ itself.

I.7. MARTINGALE REPRESENTATION OF SQUARE-INTEGRABLE MARTINGALES 47

(iii) N ∈ H2(G;Rp) with 〈N j , X,i〉 = 0 for every i = 1, . . . , , j = 1, . . . , p and MμX [ΔN |PG] = 0.

It is immediate that the decomposition (I.64) is an orthogonal decomposition according to Defini-tion I.162. The Proposition I.165, which can be seen as a generalisation of Jacod and Shiryaev [41,Lemma III.4.24], provides a framework for the well-posedness of orthogonal decompositions. Moreover,the presented form allows for embedding discrete-time orthogonal decompositions into continuous-timeorthogonal decompositions.

Notation I.163. For X ∈ H2(G;R), X ∈ H2,d(G;R) and for every i = 1, . . . ,

• X,i denotes the i−element of X.• X,c,i denotes the i−element of the continuous part of X.• X,d,i denotes the i−element of the purely discontinuous part of X.• X,i denotes the i−element of X.• H2,⊥(G, X, μX

;Rp) :=N ∈ H2(G;Rp), N i ⊥⊥ Y for every Y ∈ L2(G, X;R) ∪ K2(G, μX

;R).

Lemma I.164. Let (X, X) ∈ H2(G;R)×H2,d(G;R) such that MμX [ΔX|PG] = 0. Then, for every

Y ∈ L2(G, X;R), Y ∈ K2(G, μX

;R) holds Y ⊥⊥ Y . In particular, 〈X, X〉 = 0.

Proof. For every i = 1, · · · , and every Y ∈ K2(G, μX

;R) holds

〈X,i, Y 〉 = 〈X,c,i, Y 〉+ 〈X,d,i, Y 〉 = 0, (I.65)

where the summand 〈X,c,i, Y 〉 vanishes because X,c,i ∈ H2,c(G;R) and Y ∈ H2,d(G;R), while thesummand 〈X,d,i, Y 〉 equals to 0 in view of the assumption M

μX [ΔX|PG] = 0 and Theorem I.93. The

equality (I.65) already proves the second statement.Assume now the factorisation

〈X〉 =∫(0,·]

d〈X〉sdFs

dFs.

In view of (I.65) we obtain rY

:= d〈Y ,X,i〉sdFs

= 0. Consequently, by Theorem I.67.(v) we have for every

Z ∈ H2(G, X;R1×) that

〈Y , Z ·X〉 =∫(0,·]

Z(rY

) dFs = 0.

Proposition I.165. Let (X, X) ∈ H2(G;R) × H2,d(G;R) with MμX [ΔX|PG] = 0. For every

Y ∈ H2(G;Rp) the orthogonal decomposition of Y with respect to (G, X, μX

), say

Y = Y0 + Z ·X + U μ(X,G) +N,

is well-defined and unique up to indistinguishability. In particular, N ∈ H2,⊥(G, X, μX

;Rp).

Proof. By the Galtchouk–Kunita–Watanabe Decomposition, see Theorem I.69, there exists Z ∈H2(G, X;Rp×) such that

Y j − Y j0 = (Z ·X)j + ĎN j (I.66)

with 〈ĎN j , X,i〉 = 0 for i = 1, . . . , and j = 1, . . . , p. Moreover, by Theorem I.95, there exists a unique

U ∈ H2(G, μX

;Rp) and N ∈ H2(G;Rp) with MμX [ΔN |PG] = 0 such that

ĎN j = U j μ(X,G) +N j for every j = 1, . . . , p. (I.67)

In total, we have determined Z ∈ H2(G, X;Rp×), U ∈ H2(G, μX

;Rp) and N ∈ H2(G;Rp) such that

Y = Z ·X + U μ(X,G) +N,

which can be written without loss of generality as

Y = Y0 + Z ·X + U μ(X,G) +N. (I.68)

We have to verify that this decomposition satisfies the properties (ii)-(iii) of Definition I.162 and,moreover, that it does not depend on the way we have determined Z,U and N


We will prove initially that the G−predictable function U is the one characterised by the triplet

(G, μX

, Y ). To this end, we are going to prove that MμX

[Δ(Z ·X)

∣∣PG]= 0. By Theorem I.67.(v), we

can write

Δ(Z ·X)d,i =∑

n=1

ZinΔX,d,n for every i = 1, . . . , p.

Therefore, for every positive and bounded G−predictable function W holds

MμX

[WΔ(Z ·X)d,i

]= M

μX

[W

∑n=1

ZinΔX,d,n]=

∑n=1

MμX

[WZinΔX,d,n]

=∑

n=1

MμX

[WZinM

μX

[ΔX,d,n∣∣PG

]]= 0,

where we used the assumption MμX [ΔX|PG] = 0 and that Z is a G−predictable process in order to

conclude. The above, after using standard monotone class arguments, allows us to conclude the required

property MμX

[Δ(Z ·X)

∣∣PG]= 0. By (I.68), the equalities

MμX [Δ(Z ·X)|PG] = M

μX [ΔX|PG] = 0

and the linearity of the Doleans-Dade measure Mμ we obtain that the following hold MμX−almost

everywhere

MμX

[ΔY∣∣PG]= M

μX

[ΔĎN∣∣PG]= M

μX

[Δ(U μ(X,G))

∣∣PG].

Hence the G−predictable function U is uniquely determined MμX−almost everywhere, see Theorem I.95.

We need to prove now that 〈Z · X, U μ(X,G)〉 = 0 as well as 〈N,X〉 = 0. But the former isimmediate by Lemma I.164, since it verifies that

〈Z ·X, U μ(X,G)〉ij = 0 for every i, j = 1, . . . , p. (I.69)

We proceed to prove the 〈N,X〉 = 0. To this end let is fix an i = 1, · · · , and a j ∈ 1, . . . , p. Finally,by (I.66) and (I.69) we obtain for every i = 1, . . . ,

〈N j , X,i〉 (I.67)= 〈ĎN j , X,i〉 − 〈U j μ(X,G), X,i〉 = 0. (I.70)

To sum up,

(i) Z ∈ H2(G, X;Rp×) and U ∈ H2(G, μX

;Rp), by (I.66) and (I.67). Moreover, Z·X and Uμ(X,G)

are unique up to indistinguishability.

(ii) 〈Z ·X, U μ(X,G)〉 = 0 by (I.69).(iii) 〈N j , X,i〉 = 0 for every j = 1, . . . , p, i = 1, . . . , and M

μX [ΔN |PG] = 0, by (I.70) and (I.67)

respectively.

In view of the above properties, we can argue analogously to Lemma I.164 in order to prove that 〈N,Y 〉 =0 for every Y ∈ L2(G, X;Rp) and that 〈N,Y 〉 = 0 for every Y ∈ K2(G, μX

;Rp). Therefore, N ∈H2,⊥(G, X, μX

;Rp).

In view of the framework of Proposition I.165, we will work

• In Chapter II with a process ĎX, where ĎX := (X, X) ∈ H2(G;R) × H2,d(G;R) such that

MμX

[ΔX∣∣PG

]= 0.

• In Section III.8 and Chapter IV we will work with a sequence (ĎXk)k∈N, where for every k ∈ N

ĎXk := (Xk,, Xk,) ∈ H2(G;R) × H2,d(G;R) such that MμXk,

[ΔXk,∣∣PG

k]= 0 and the probability

space will be endowed with a sequence of filtrations (Gk)k∈N.

However, we will need also a process CĎX ∈ V+

pred(G;R), resp. a sequence (CĎXk

) ⊂ V+pred(G;R), which

will be responsible, among others, for the factorisation of 〈X〉, resp. (〈Xk,〉)k∈N.

In the following lemma, which is essentially Lemma I.166, we prove that for ĎX := (X, X) as in

Lemma I.164 there exists a process CĎX with the desired properties. For this reason, we collect in the

assumption following the lemma the properties we will need for the pair (ĎX,CĎX) to satisfy.

I.7. MARTINGALE REPRESENTATION OF SQUARE-INTEGRABLE MARTINGALES 49

Lemma I.166. Let ĎX := (X, X) ∈ H2(G;R) × H2,d(G;R) with MμXd [ΔXc|PG] = 0. Then, there

exists CĎX ∈ V+

pred(G;R) such that

(i) Each component of 〈X〉 is absolutely continuous with respect to CĎX . In other words, there exists

a predictable, positive definite and symmetric × −matrix d〈X〉dC ĎX such that for any 1 ≤ i, j ≤

〈X〉ij· =

∫(0,·]

d〈X〉ijsdCĎX

s

dCĎXs .

(ii) The disintegration property given CĎX holds for the compensator ν(X

,G), i.e. there exists a tran-

sition kernel KĎX : (Ω × R+,P) −→ R(R,B(R)), where R(R,B(R)

)is the space of Radon measures

on(R,B(R)

)such that

ν(X,G)(ω; dt, dx) = K

ĎXt (ω; dx) dC

ĎXt .

Moreover, for every every pair (ω, t) ∈ Ω× R+ the transition kernel possesses the following properties

KĎXt (ω; 0) = 0,

∫R

(‖x‖22 ∧ 1)KĎXt (ω; dx) ≤ 1 and ΔC

ĎX(ω)K

ĎXt (ω;R) ≤ 1. (I.71)

(iii) CĎX can be chosen to be continuous if and only if ĎX is G−quasi-left-continuous.

Proof. We can follow exactly the same arguments as in Jacod and Shiryaev [41, Proposition II.2.9]for the process

CĎX· :=

∥∥Var(〈X〉)·∥∥1 + (‖Id‖22 ∧ 1) ∗ ν(X,G)· .

We need to underline that under our framework the process 〈X〉 is not necessarily continuous. However,Lemma I.164 verifies that we can indeed follow the same arguments as in [41, Proposition II.2.9].

Assumption (C). We will say that the pair (ĎX,CĎX) satisfies Assumption C under G, where ĎX :=

(X, X), if:

(i) ĎX ∈ H2(G;R)× H2,d(G;R) with MμX [ΔX|PG] = 0.

(ii) CĎX ∈ V+

pred(G;R).

(iii) The process CĎX satisfies the properties (i) - (iii) of Lemma I.166.

Remark I.167. Given an X ∈ H2(G;R) we have that its natural pair (Xc, Xd) satisfies trivially the

condition MμXd [ΔXc|PG] = 0, since ΔXc = 0 identically. Let us denote ĎX := (Xc, Xd) and mention

that there exist several possible choices for CĎX such that the pair (ĎX,C

ĎX) satisfies Assumption (C). InJacod and Shiryaev [41, Proposition II.2.9], for example, the following process is used∥∥Var(〈Xc,i, Xc,j〉)·‖1 + (‖Id‖22 ∧ 1) ∗ ν(Xd,G)

· , (I.72)

while one could also take

Tr[〈Xc〉·] + ‖Id‖22 ∗ ν(Xd,G)· .

CHAPTER II

Backward Stochastic Differential Equations with Jumps

II.1. Introduction

This chapter, which follows closely Papapantoleon et al. [54], is devoted to obtaining a wellposednessresult for multidimensional backward stochastic differential equations with jumps1 driven by a square-integrable martingale in a filtration which may be stochastically discontinuous. Initially, let us explainroughly what is a BSDE.2 To this end let us set the framework. The probability space (Ω,G,P) is given andit will be assumed fixed throughout the chapter. Assume, now, that we are also given an arbitrary sextuple(G, T,X, ξ, C, f), where G is a filtration, T is a G−stopping time which may be infinite, XT ∈ H2(G;R`),ξ ∈ L2(FT ;Rp), C ∈ V+

pred(G;R) and f is a stochastic operator which will be called the generator of

the BSDE; the properties the data should satisfy will be made precise in (F1) - (F5) on p. 58. We areinterested in proving the existence and the uniqueness of a quadruple (Y,Z, U,N) which satisfies on J0, T K

Yt = Y0 −∫

(0,t]

f(s, Ys, Zs, U(s, ·))dCs

−∫ t

0

ZsdXcs −

∫ t

0

∫R`U(s, x)µ(ds,dx)−

∫ t

0

dNs with YT = ξ P− a.s.,

or equivalently

Yt = ξ+

∫(t,T ]

f(s, Ys, Zs, U(s, ·))dCs−∫ T

t

ZsdXcs −∫ T

t


∫ T

t

dNs, P−a.s.. (II.1)

The quadruple (Y,Z, U,N) will be said to be the solution of the BSDE (II.1) or simply the solutionof the BSDE.

In general, the existence or the uniqueness of the solution, hereinafter (E/U), is not guaranteed for anarbitrarily given sextuple as above. It turns out that for the (E/U) the properties of the Lebesgue–Stieltjesintegrator C play a critical role. More precisely, when C is continuous and the data satisfy a suitableL2−type condition, then the a priori estimates of the BSDE can be obtained, see e.g. El Karoui andHuang [29] and El Karoui and Mazliak [30, Subsection 2.1.2] for the need of the integrability condition.However, when C is allowed to have jumps, i.e. when the driving martingale is not G−quasi-left-continuous, the counter-example of Confortola, Fuhrman, and Jacod [22, Section 4.3] shows that thereare cases that either the solution does not exist or there are infinitely many solutions. This interestingphenomenon is further analysed in Section II.2. Then, we proceed to show that a framework for (E/U)finally can be provided once the stochastic operator f satisfies a stochastic Lipschitz condition and theinterplay between the stochastic bounds of f and the jumps of the integrator C can be tamed in thesense of Condition (II.18). This is stated in Theorem II.14, where the (E/U) is ensured in appropriatelyweighted spaces. Our result allows in particular to obtain a wellposedness result for BSDEs driven bydiscrete-time approximations of square-integrable martingales.

It would be an unforgivable omission if we had not mentioned that it was the seminal work of Pardouxand Peng [55] that established the theory of non-linear BSDEs as a notably active field of research. Sincewe will deal with a Lipschitz-type generator we intend to mention only the related works, although thetheory of BSDEs has also been developed towards other directions. However, their abundance does notallow us to claim that the following list is comprehensive; to the contrary, we could regard it as rathersuperficial.

1Hereinafter, we will refer to a backward stochastic differential equation with jumps as BSDE. When we refer to manysuch equations the acronym alters to BSDEs.

2For a detailed introduction in the case the BSDE is driven by a Brownian motion, the reader may consult El Karoui,

Peng, and Quenez [31].

51

52 II. BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS

We have initially to go back some decades, more precisely in 1973, when Davis and Varaiya [24] andBismut [8, 9]3 have dealt with linear versions of BSDEs driven by a Brownian motion. Then we proceedto 1990, when the first systematic treatment of non-linear BSDEs was made by Pardoux and Peng [55]in the case the driving martingale is the Brownian motion and the generator is globally Lipschitz. Moregeneral results on BSDEs driven by a cadlag martingale were obtained by Buckdahn [16], El Karouiet al. [31], as well as El Karoui and Huang [29] and Carbone, Ferrario, and Santacroce [17], where theunderlying filtration is assumed to be quasi-left-continuous. To the best of our knowledge, the firstarticles that went beyond this assumption were developed in a very nice series of papers by Cohen andElliott [19] and Cohen, Elliott, and Pearce [21], where the only assumption on the filtration is thatthe associated L2−space is separable, so that a very general martingale representation result due toDavis and Varaiya [25], involving countably many orthogonal martingales, holds. In these works, themartingales driving the BSDE are actually imposed by the filtration, and not chosen a priori, and thenon-decreasing process C is not necessarily related to them, but has to be deterministic and can havejumps in general, though they have to be small for existence to hold (see [19, Theorem 5.1]). Similarly,Bandini [6] obtained wellposedness results in a context of a general filtration allowing for jumps, with afixed driving martingale and associated random process C, which must have again small jumps, see [6,Equation (1.1)]. We mention again the work by Confortola et al. [22] which concentrates on the pure-jump general case and gives in particular counterexamples to existence. Finally, Bouchard, Possamaı,Tan, and Zhou [13] provide a general method to obtain a priori estimates in general filtrations whenthe martingale driving the equation has quadratic variation absolutely continuous with respect to theLebesgue measure.

We describe, now, the structure of the chapter. Section II.2, which is mainly based on the counterex-ample of Confortola et al. [22, Section 4.3], intends to shed some light on the various situations in whicha solution may or may not exist. In Section II.3 we provide the main notation we are going to use in thechapter, which will be used in Chapter IV as well. Then, in Section II.4 we state the theorem for (E/U)of (II.1). The a priori estimates for (II.1) are provided in Section II.5, while in the subsequent sectionthe proof of the theorem is presented. In Section II.7 we compare our results with the related literature.

II.2. A simple counterexample and its analysis

As we have already mentioned, Confortola et al. [22, Section 4.3 Case (1)] provided a counterexampleto the existence or the uniqueness of the solution of a BSDE in case the integrator C is not a continuousprocess. We are going to have a detailed look on their counterexample, and, based on it, we are going todiscuss the well-posedness of variants of (II.1). We will see that the well-posedness of the BSDE is affectedby seemingly slight changes in its form. To this end, let us provide the framework of the section,whichwill be set to be as simple as possible.

The probability space (Ω,G,P) has been already given. We assume that there exists a Π ∈ G suchthat p := P(Π) ∈ (0, 1). We define, now, for r ∈ (0,∞) the random variable ρ by

ρ(ω) =

r , when ω ∈ Π,

∞, for ω ∈ Πc.

We proceed by defining the one-point process X = 1[ρ,∞), whose natural filtration is denoted by FX . In

other words, ρ is an FX−stopping time. By Jacod and Shiryaev [41, Lemma III.1.29] we have a char-acterisation of the elements of FX . Moreover, by the aforementioned lemma we have a characterisationof the FX−predictable functions, which in this special case are processes. We have, then, that U isFX−predictable if and only if is of the form

U(ω, t) = U0 + U∞1(r ,∞](t)1Π(ω), (II.2)

where U0 is FX0 −measurable and U∞ is FXr −measurable. The compensator ofX under FX is described byJacod and Shiryaev [41, Theorem III.1.33]. Indeed, for Pρ being the distribution of ρ on

([0,∞],B([0,∞])

),

we calculate initially the distribution function Fρ of ρ, which is given by

Fρ(t) =

0, for t ∈ [0, r ),

p, for t ∈ [r ,∞),

1, for t =∞and also Pρ([t,∞]) =

1, for t ∈ [0, r ],

1− p, for t ∈ (r ,∞].

3The published version of [24] states that the article was received on October 27, 1971. It is also present in bibliography

of [9], though it is never referred to in the text.

II.2. A SIMPLE COUNTEREXAMPLE AND ITS ANALYSIS 53

Consequently, the cadlag function

Ct :=

∫(0,t]

1

Pρ([s,∞]

)Fρ(ds) =

0, for t ∈ (0, r ),

∆Fρ(r)Pρ([r ,∞]) = p, for t ∈ [r ,∞),

∆Fρ(r)Pρ([r ,∞]) +

∆Fρ(∞)Pρ(∞) = p+ 1, for t =∞

restricted on R+ is the FX -compensator of the processX, i.e. the process X := X−C is an FX−martingaleon R+.

We proceed to designate the rest of the data. We fix some generator f : [0, T ] × R × R −→ R, theterminal time T ∈ (r ,∞) and the terminal condition ξ. Now we consider the following BSDEs

Yt +

∫ T

t

UsµX(ds) = ξ +

∫(t,T ]

f(s, Ys−, Us)dCs, (II.3)

Yt +

∫ T

t

UsµX(ds) = ξ +

∫(t,T ]

f(s, Ys−, Us)dCs, (II.4)

Yt +

∫ T

t

UsµX(ds) = ξ +

∫(t,T ]

f(s, Ys, Us)dCs (II.5)

and

Yt +

∫ T

t

UsµX(ds) = ξ +

∫(t,T ]

f(s, Ys, Us)dCs. (II.6)

The first case corresponds to the one Confortola et al. [22] considered, where they chose the generatorto have the form f(t, y, u) = 1

p (y + g(u)) for a deterministic function g : R −→ R. The authors proved

that the BSDE (II.3) can admit either infinitely many solutions or none. We would like to examineapart from the aforementioned counterexample also the BSDEs (II.4) - (II.6), which can be categorisedas follows. The BSDE (II.4) is a modification of (II.3), where the stochastic integral is assumed to betaken with respect to the compensated integer-valued random measure. The BSDEs (II.5) and (II.6) aremodifications of the first two, in the sense that now the dependence of the Lebesgue–Stieltjes integral ison Y instead of Y−.

Before we proceed, let us comment on the structure of the solution we are seeking. Given the structureof the filtration FX , the terminal condition ξ can always have a decomposition of the form

ξ(ω) =: ξΠ1Π(ω) + ξΠc1Πc(ω), (ξΠ, ξΠc) ∈ R× R.

Once again because of the structure of FX and of the form (II.2) for the FX−predictable functions, thepossible solutions for the BSDEs above necessarily have the following form

Yt(ω) = Y01[0,r)(t) + ξΠ1[r ,∞)(t)1Π(ω) + ξΠc1[r ,∞)(t)1Πc(ω),

and

Ut(ω) = U0 + U∞1(r ,∞](t)1Π(ω),

for some Y0, U0 ∈ R and U∞ is FXr −measurable. However, only the value Ur is actually involved becauseof the form of X and C. Now we can distinguish between the following cases:

(C1) Consider the BSDE (II.3). Then, there exists a solution if and only if there exists a fixed point,called Y ?0 , for the equation

ξΠ + pf(r , x, ξΠ − ξΠc) = x.

The pair (Y ?0 , ξΠ − ξΠc) is a solution of (II.3). The solution is unique if and only if the fixed point is

unique. In case f is globally Lipschitz with respect to its second argument, i.e.

|f(t, y1, u)− f(t, y2, u)| ≤ Ly|y1 − y2|,then, a sufficient condition for the existence and uniqueness of the solution is Ly∆Cr < 1.

(C2) Consider the BSDE (II.4), where, comparing to the previous case, the integrand of the Lebesgue–Stieltjes integral depends on Y instead of Y−. Then, there exists a solution if and only if there exists afixed point, called Y ?0 , for the equation

ξΠ + pf(r , x, ξΠ − ξΠc) + p(ξΠ − ξΠc) = x.


The pair (Y 0 , ξ

Π − ξΠc

) is a solution of (II.4). The solution is unique if and only if the fixed point isunique. In case f is globally Lipschitz with respect to the second argument as above, a sufficient conditionfor the existence and uniqueness of the solution is again LyΔCr < 1.

(C3) Consider now the BSDE (II.5). Then, there exists a solution if and only if there exists a fixedpoint, called υ(r ), for the equation

ξΠ − ξΠc − pf

(r , ξΠ

c

, x)+ pf(r , ξΠ, x) = x.

The pair (ξΠc

+pf(

r , ξΠc

+pυ(r ), υ(r )), υ(r )) is a solution of (II.5). The solution is unique if and only

if the fixed point is unique. In case f is globally Lipschitz with respect to its third argument, i.e.

|f(t, y, u1)− f(t, y, u2)| ≤ Lu|u1 − u2|,then, a sufficient condition for the existence and uniqueness of the solution is 2LuΔCr < 1. This conditionis not necessary: let f ′(t, y, u) = 1

p (g(y) + u), where g is a deterministic function, then it holds that

LuΔCr = 1; however, (II.5) admits a unique solution, which is given by the pair(ξΠ + g(ξΠ), ξΠ − ξΠ

c

+ g(ξΠ)− g(ξΠc

)).

(C4) Finally, consider a BSDE similar to (II.5), where the stochastic integral is taken with respect tothe compensated jump process. Then, there exists a solution if and only if there exists a fixed point,called υ(r ), for the equation

ξΠ − ξΠc − pf

(r , ξΠ

c

, x)+ pf(r , ξΠ, x) = x.

The pair (ξΠc

+ pf(

r , ξΠc

, υ(r )), υ(r )) is a solution of (II.6). The solution is unique if and only if the

fixed point is unique. In case f is globally Lipschitz with respect to its third argument as above, asufficient condition for the existence and uniqueness of the solution is again 2LuΔCr < 1. Once againthis condition in not necessary; indeed, for f ′ as in (C3), LuΔCr = 1, while the unique solution of theBSDE (II.6) is the pair(

(1− p)[ξΠ + g(ξΠ)] + p[ξΠc

+ g(ξΠc

)], ξΠ − ξΠc

+ g(ξΠ)− g(ξΠc

)).

Now, returning to the original counterexample of [22], we can observe that the sufficient conditionLyΔCr < 1 is violated there, which explains why wellposedness issues can arise. However, an importantobservation here is that the structure of the generator plays a crucial role as well. Indeed, if we considerthe same BSDE with the following generator f(t, y, u) = (y + g(u)) with = 1

p , then the BSDE admits

a unique solution.

Remark II.1. We observe that the dependence of the integrand on Y or Y− is not always that innocuous.Indeed, the same BSDE might have a solution in the one formulation but not in the other. Observefurthermore that in the first situation the Lipschitz constant Ly appears in the condition for the existenceand uniqueness of a solution, while in the second case the Lipschitz constant Lu appears. We will seethat in our framework we can treat both cases simultaneously, hence naturally both Lipschitz constantswill appear in our condition; see Condition (F4) and Conditions (II.19).

II.3. Suitable spaces and associated results

We start by endowing the probability space (Ω,G,P) with an arbitrary filtration G. Moreover, we

consider an arbitrary ĎX := (X, X) ∈ H2(G;R) × H2,d(G;R) and an arbitrary CĎX ∈ V+(G;R) such

that the pair (ĎX,CĎX) satisfies Assumption (C) under G.

Notation II.2. To the pair (ĎX,CĎX) which satisfies Assumption (C) under G (with associated transition

kernel KĎX) we will associate the following:

• cĎX denotes the square root of the Radon–Nikodym derivative d〈X〉/dCĎX . In other words, c

ĎX isthe unique G−predictable, positive definite, symmetric and R×−valued process for which

(cĎX)cĎX =

d〈X〉dCĎX

. (II.7)

• μĎXΔ denotes the random measure on (R+,B(R+)) which is defined via

μĎXΔ (ω; (0, ·]) :=

∑0<s≤·

(ΔC

ĎXs (ω)

)2and for which, moreover, holds

dμĎXΔ

dCĎX(ω; ·) = ΔC

ĎX· (ω). (II.8)

II.3. SUITABLE SPACES AND ASSOCIATED RESULTS 55

• For W : Ω −−→ Rp such that W (ω, s, ·) :(R,B(R)

) −−→ (Rp,B(Rp)

)is measurable for dP ⊗

dCĎX−almost every (ω, s) ∈ Ω× R+, we abuse notation slightly and we denote

KĎXt (Ws(·))(ω) :=

∫R

W (ω, s, x)KĎXt (ω; dx), s, t ≥ 0.

• For W : Ω −−→ Rp such that W (ω, s, ·) :(R,B(R)

) −−→ (Rp,B(Rp)

)is measurable for dP ⊗

dCĎX−almost every (ω, s) ∈ Ω× R+ we denote(|||Ws(·)|||ĎX

t (ω))2

:= KĎXt (‖Ws(·)‖22)(ω)−ΔC

ĎXt (ω)‖KĎX

t (Ws(·))(ω)‖22 for every (ω, t) ∈ Ω× R+. (II.9)

Lemma II.3. To the pair (ĎX,CĎX), which satisfies the Assumption (C) under G, we associate the process

〈ĎX〉 := 〈X +X〉. Then,

〈ĎX〉· =∫(0,·]

[c

ĎXs c

ĎXs + K

ĎXs (q)−ΔC

ĎXs K

ĎXs (Id )K

ĎXs (Id)

]dC

ĎXs .

Proof. Since the pair (ĎX,CĎX) satisfies the Assumption (C) under G, we have in particular that

〈ĎX〉 = 〈X〉+ 2〈X, X〉+ 〈X〉 Lem. I.164= 〈X〉+ 〈X〉. (II.10)

Therefore, the process 〈ĎX〉 admits the following representation

〈ĎX〉· := 〈X〉· + 〈X〉·=

∫(0,·]

d〈X〉sdCĎX

s

dCĎXs + q ∗ ν(X,G)

· −∑

0<s≤·

∫R

xν(X,G)(s × dx)

∫R

xν(X,G)(s × dx)

=

∫(0,·]

[(c

ĎXs )cĎX

s +

∫R

xxKĎXs (dx)

]dC

ĎXs −

∑0<s≤·

∫R

xKĎXs (dx)ΔC

ĎXs

∫R

xKĎXs (dx)ΔC

ĎXs

=

∫(0,·]

[(c

ĎXs )cĎX

s + KĎXs (q)

]dC

ĎXs −

∑s≤·

KĎXs (Id )K

ĎXs (Id)

(ΔC

ĎXs

)2=

∫(0,·]

[(c

ĎXs )cĎX

s + KĎXs (q)−ΔC

ĎXs K

ĎXs (Id )K

ĎXs (Id)

]dC

ĎXs .

We proceed now by defining the spaces that will be necessary for our analysis, see also El Karoui andHuang [29]. Let β ≥ 0, A : (Ω×R+,G ⊗ B(R+)) −→ R+ be a cadlag, increasing and measurable processand τ be a G−stopping time.

Notation II.4. We define the following spaces:

• L2A,β(Gτ ;R

p) :=ξ, Rp−valued, Gτ−measurable, ‖ξ‖2

L2A,β(Gτ ;Rp) := E

[eβAτ ‖ξ‖22

]< ∞

,

• H2A,β(G;Rp) :=

M ∈ H2(G;Rp), ‖M‖2H2

A,β(G;Rp) := E

[ ∫(0,τ ]

eβAtdTr [〈M〉t]]< ∞

,

• H2A,β(G, ĎX;Rp) :=

φ is an Rp−valued G−optional semimartingale with cadlag paths and

‖φ‖2H2

A,β(G,ĎX;Rp) := E

[ ∫(0,τ ]

eβAt‖φt‖2dCĎXt

]< ∞

,

• S2A,β(G, ĎX;Rp) :=

φ is an Rp−valued G−optional semimartingale with cadlag paths and

‖φ‖2S2A,β(G,ĎX;Rp) := E

[sup

t∈0,τ

eβAt‖φt‖2]< ∞

,

• H2A,β(G, X;Rp×) :=

Z ∈ H2(G, X;Rp×), ‖Z‖2

H2A,β(G,X;Rp×) < ∞

with ‖Z‖2

H2A,β(G,X;Rp×) := E

[ ∫(0,τ ]

eβAtdTr[ 〈Z ·X〉t]],


• H2A,β(G, μX

;Rp) :=

U ∈ H2(G, X;Rp

), ‖U‖H2

A,β(G,μX;Rp)

< ∞,

with ‖U‖H2

A,β(G,μX;Rp)

:= E

[ ∫(0,τ ]

eβAtdTr[〈U μ(X,G)〉t]],

• H2,⊥A,β(G, X, μX

;Rp) :=

M ∈ H2,⊥(G, X, μX

;Rp), ‖M‖2H2,⊥A,β(G,X,μX

;Rp)< ∞

with ‖M‖2H2,⊥

A,β(G,X,μX;Rp)

:= E

[ ∫(0,τ ]

eβAtdTr[ 〈M〉t]].

• For (Y, Z, U,N) ∈ H2A,β(G, ĎX;Rp)×H2

A,β(G, X;Rp×)×H2A,β(G, μX

;Rp)× H2,⊥A,β(G, X, μX

;Rp)

and assuming that A = α2 · CĎX for a measurable process α : (Ω× R+,G ⊗ B(R+)) −→ R, we define

‖(Y, Z, U,N)‖2A,β,ĎX :=

‖αY ‖2H2

A,β(G,ĎX;Rp) + ‖Z‖2H2

A,β(G,X;Rp×) + ‖U‖2H2

A,β(G,μX;Rp)

+ ‖N‖2H2,⊥A,β(G,X,μX

;Rp).

• For (Y, Z, U,N) ∈ S2A,β(G, ĎX;Rp)×H2

A,β(G, X;Rp×)×H2A,β(G, μX

;Rp)× H2,⊥A,β(G, X, μX

;Rp)

and assuming that A = α2 · CĎX for a measurable process α : (Ω× R+,G ⊗ B(R+)) −→ R, we define

‖(Y, Z, U,N)‖2,A,β,ĎX :=

‖Y ‖2S2A,β(G,ĎX;Rp) + ‖Z‖2

H2A,β(G,X;Rp×) + ‖U‖2

H2A,β(G,μX

;Rp)+ ‖N‖2H2,⊥

A,β(G,X,μX;Rp)

.

The next lemma will be useful for future computations and, in addition, justifies the definition of thenorms on the spaces provided above.

Lemma II.5. Let (Z,U) ∈ H2A,β(G, X;Rp×)×H2

A,β(G, μX

;Rp). Then

‖Z‖2H2

A,β(G,X;Rp×) = E

[∫(0,τ ]

eβAt‖ctZt‖22 dCĎXt

](II.11)

and

‖U‖2H2

A,β(G,μX;Rp)

= E

[∫(0,τ ]

eβAt( |||Ut(·)|||ĎX

t

)2dC

ĎXt

]. (II.12)

Furthermore |||Ut(·)|||ĎXt ≥ 0, for every t ≥ 0 and

‖Z ·X + U μ(X,G)‖2H2A,β(G;Rp) = ‖Z‖H2

A,β(G,X;Rp×) + ‖U‖H2

A,β(G,μX;Rp)

.

Proof. Let Z := (Z1, . . . , Zp) ∈ H2A,β(G, X;Rp×), then using Theorem I.67, we get for i, j =

1, . . . , p

〈Z ·X〉ij =∫(0,·]

Zis

d〈X〉sdCĎX

s

(Zjs)

dCĎXs =

∫(0,·]

Zisc

ĎXs (c

ĎXs )(Zj

s) dC

ĎXs , (II.13)

for cĎX as introduced in (II.7). The first result is then obvious in view of Notation I.66 and Notation II.4.

Using Assumption (C) and following Notation II.2, we get for U a G−predictable function takingvalues in Rp that ∫

R

U(ω, t, x) ν(X,G)(ω; t × dx) = K

ĎXt (Ut(·))(ω)ΔC

ĎXt (ω), t ≥ 0,

and ∫R

‖U(ω, t, x)‖22 ν(X,G)(ω; t × dx) =

∫R

‖U(ω, t, x)‖22KĎXt (ω; dx)ΔC

ĎXt (ω)

= KĎXt (‖Ut(·)‖22)(ω)ΔC

ĎXt (ω), t ≥ 0.

In particular, now, for U ∈ H2A,β(G, μX

;Rp) and similarly to the computations in Lemma II.3, we have

〈U μ(X,G)〉· =∫(0,·]

[K

ĎXt

((U

t Ut)(·))−ΔC

ĎXs K

ĎXt

(Ut(·)

)K

ĎXt (Ut(·))

]dC

ĎXs , (II.14)

II.4. FRAMEWORK AND STATEMENT OF THE THEOREM 57

from which the second result is also clear in view of Notation II.4. Moreover, we have

‖Z ·X + U μ(X,G)‖2H2A,β(G;Rp) = E

[∫(0,τ ]

eβAtdTr[〈Z ·X + U μ(X,G)〉t

]]

= E

[∫(0,τ ]

eβAtdTr [ 〈Z ·X〉t] +∫(0,τ ]

eβAtdTr[〈U μ(X,G)〉t

]],

where the second equality holds because 〈Z · X, U μ(X,G)〉 = 0. This is true by Lemma I.164, since

the pair (ĎX,CĎX) satisfies Assumption (C) under G.

Notice finally that the process Tr[ 〈U μ(X,G)〉] is non-decreasing, and observe that

ΔTr[〈U μ(X,G)〉t

]=(|||Ut(·)|||ĎX

t

)2ΔC

ĎXt , t ≥ 0. (II.15)

Since CĎX is non-decreasing, we can deduce that |||Ut(·)|||ĎX

t ≥ 0. We conclude this section with the following convenient result. We provide initially the necessary

notation.

Notation II.6. • For dP⊗ dCĎX − a.e. (ω, t) ∈ Ω× R+ we denote

HĎXω,t :=

U : (R,B(R)) −→ (Rp,B(Rp)), ||| U(·)|||ĎX

t (ω) < ∞.

• HĎX :=

U : [0, T ]× Ω× R −→ Rp, Ut(·) ∈ H

ĎXω,t, for dP⊗ dC

ĎX − a.e. (ω, t) ∈ Ω× R+

.

Lemma II.7. The space(H

ĎXω,t, |||·|||

ĎXt (ω)

)is Polish, for dP⊗ dC

ĎX − a.e. (ω, t) ∈ Ω× R+.

Proof. The fact that |||·|||ĎXt (ω) is indeed a norm is immediate from (II.15) and Kunita–Watanabe’s

inequality. Then, the space is clearly Polish since Kt(ω; dx) is a regular measure, as it integrates x −→‖x‖22 for dP⊗ dC

ĎX − a.e. (ω, t) ∈ Ω× R+.

II.4. Framework and statement of the theorem

In this section we fix a filtration G, a G−stopping time T , a process ĎX := (X, X) and a CĎX ∈

V+(G;R) such that

the pair of stopped processes (ĎX T , (CĎX)T ) satisfies Assumption (C) under G.

We fix also a positive integer n, which denotes the dimension of the state space of the solution we areseeking, that is the state space will be Rn. In order to ease notation, since T is given, we will refer to thestopped processes ĎX T , (X)T , (X)T ,(C

ĎX)T simply as ĎX, X, X and CĎX . Hence, the time interval on

which we will be working throughout this section will always be the stochastic time interval 0, T . Inaddition, a non-decreasing process A will be fixed below, see (II.17). In order to further simplify notation,and since there is no danger of confusion, we will omit G, ĎX,A and the state spaces, from the spaces andthe norms.

Notation II.8. For any β ≥ 0 and (ω, t) ∈ Ω× R+ we define:

• P := PG, P := PG.• L2

β := L2β(GT ;R

n).

• H2β := H2

A,β(G, ĎX;Rn), H2,β := H2

β(G, X;Rn×), H2,β := H2

β(G, μX

;Rn).

• S2β := S2

A,β(G, ĎX;Rn), H2,⊥β := H2,⊥

A,β(G, X, μX

;Rn), H2β := H2

A,β(G;Rn).

• |||·|||t := |||·|||ĎXt , Hω,t := H

ĎXω,t, H := H

ĎX .• ‖ · ‖β := ‖ · ‖A,β,ĎX , ‖ · ‖,β := ‖ · ‖,A,β,ĎX .

• C := CĎX , c := c

ĎX , μ := μX

, ν := ν(X,G), μ := μ(X,G).

When β = 0, we also suppress it from the notation of the previous spaces.

The data (G, T, ĎX, ξ, C, f) of the BSDE

Yt = ξ+

∫(t,T ]

f(s, Ys, Zs, U(s, ·))dCs−∫ T

t

ZsdXcs −∫ T

t

∫R

U(s, x)μ(ds, dx)−∫ T

t

dNs, P−a.s.. (II.1)


should satisfy the following conditions:

(F1)4 The process ĎX = (X, X) belongs to H2(G;R) × H2,d(G;R) and the pair (ĎX,C) satisfiesAssumption (C).

(F2) The terminal condition satisfies ξ ∈ L2βfor some β > 0.

(F3) The generator5 of the equation f : Ω × R+ × Rn × Rn× × H −→ Rn is such that for any(y, z, u) ∈ Rn × Rn× × H, the map

(ω, t) −→ f(ω, t, y, z, u(ω, t, ·)) is Ft ⊗ B([0, t])− measurable.

Moreover, f satisfies a stochastic Lipschitz condition, that is to say there exist

r : (Ω× R+,P) −→ (R+,B(R+)) and ϑ = (θc, θd) : (Ω× R+,P) −→ (R2+,B(R2

+)),

such that, for dP⊗ dC − a.e. (ω, t) ∈ Ω× R+

‖f(ω, t, y, z, u(ω, t, ·))− f(ω, t, y′, z′, u′(ω, t, ·))‖22≤ rt(ω)‖y − y′‖22 + θct (ω)‖ct(ω)(z − z′)‖22 + θdt (ω)

(|||ut(·)− u′t(·)|||t (ω)

)2. (II.16)

(F4) Let6 α2· := max√r·, θc· , θ

d· and define the increasing, G−predictable and cadlag process

A· :=∫(0,·]

α2s dCs. (II.17)

Then there exists Φ > 0 such that

ΔAt(ω) ≤ Φ, for dP⊗ dC − a.e. (ω, t) ∈ Ω× R+. (II.18)

(F5) We have for the same β as in (F2)

E

[∫(0,T ]

eβAt‖f(t, 0, 0,0)‖22

α2t

dCt

]< ∞,

where 0 denotes the null application from R to R.

Remark II.9. In the case where the integrator C of the Lebesgue–Stieltjes integral is a continuousprocess, we can choose between the integrands(

f(t, Yt, Zt, U(t, ·)))t∈0,T

and(f(t, Yt−, Zt, U(t, ·)))

t∈0,T ,

and we still obtain the same solution, as they coincide on a dP⊗dC−null set. However, in the case wherethe integrator C is cadlag, the corresponding solutions may differ. In the formulation of the problemwe have chosen the first one, while all our results can readily be adapted to the second case as well.Recall, however, that in Section II.2 we have seen that, in special cases, the conditions for existence anduniqueness of a solution in these two cases can differ significantly.

In classical results on BSDEs, the pair (ξ, f) is called standard data. In our case, the followingdefinition generalises this term.

Definition II.10. We will say that the sextuple (G, T, ĎX, ξ, C, f) is the standard data under β, whenever

its elements satisfy Assumptions (F1)–(F5) for this specific β.

Definition II.11. A solution of the BSDE (II.1) with standard data (G, T, ĎX, ξ, C, f) under β > 0 is aquadruple of processes

(Y, Z, U,N) ∈ H2β ×H

2,β ×H

2,β × H2,⊥

β or (Y, Z, U,N) ∈ S2β ×H

2,β ×H

2,β × H2,⊥

β ,

for some β ≤ β such that, P− a.s., for any t ∈ 0, T ,

Yt = ξ +

∫(t,T ]

f(s, Ys, Zs, U(s, ·))dCs −∫ T

t

Zs dXc

s −∫ T

t

∫Rn

U(s, x)μ(ds, dx)−∫ T

t

dNs. (II.1)

4We simply restate the black square above in order to have a convenient reference for it.5This is also called driver of the BSDE.6We assume, without loss of generality, that αt > 0, dP⊗ dC − a.e.

II.4. FRAMEWORK AND STATEMENT OF THE THEOREM 59

Remark II.12. We emphasise that in (II.1), the stochastic integrals are well defined since (Z,U,N) ∈H

2,β ×H

2,β × H2,⊥

β . Let us verify that the integral∫(0,·]

f(s, Ys, Zs, U(s, ·))dCs

is also well-defined. First of all, we know by definition that for any (y, z, u) ∈ Rn×Rn××H, there existsa dP⊗ dC−null set N y,z,u such that for any (ω, t) /∈ N y,z,u

f(ω, t, y, z, u(ω, t, ·)) is well defined and u(ω, t, ·) ∈ Hω,t.

Moreover, by Lemma II.7, we know also that for some dP⊗dC−null set N , we have for every (ω, t) /∈ N ,that Hω,t is Polish for the norm |||·|||t (ω), so that it admits a countable dense subset which we denote byHω,t. Let us then define

H :=u ∈ H, u(ω, t, ·) ∈ Hω,t, ∀(ω, t) /∈ N

, N :=

⋃N y,z,u, (y, z, u) ∈ Qd ×Qn× ×H,

where Q and Qn× are the subsets of R and Rn× with rational components.

Then, since H is countable, N is still a dP ⊗ dC−null set. Then, it suffices to use (F3) to re-

alize that for any (ω, t) /∈ N ∪ N , f is continuous in (y, z, u), and conclude that we can actually de-fine f(ω, t, y, z, u(ω, t, ·)) outside a universal dP ⊗ dC−null set. This implies in particular that for any

(Y, Z, U) ∈ H2β ×H

2,β ×H

2,β

f(ω, t, Yt(ω), Zt(ω), U(ω, t, ·)) is defined for dP⊗ dC − a.e. (ω, t) ∈ Ω× 0, T .

Finally, it suffices to use (F3) and (F5) to conclude that∫(0,T ]

‖f(ω, t, Yt(ω), Zt(ω), U(ω, t, ·))‖2dCt(ω) is finite P− a.s..

II.4.1. Existence and uniqueness: statement. We devote this subsection to the statement ofour theorem. Before that, we need some preliminary results of a purely analytical nature, whose proofsare relegated to Appendix A.1.

Lemma II.13. Fix β,Φ > 0 and consider the set Cβ := (γ, δ) ∈ (0, β]2, γ < δ. We define the followingquantity

ΠΦ(γ, δ) :=9

δ+ (2 + 9δ)

e(δ−γ)Φ

γ(δ − γ).

Then, the infimum of ΠΦ is given by

MΦ(β) := inf(γ,δ)∈Cβ

ΠΦ(γ, δ) =9

β+

Φ2(2 + 9β)√β2φ2 + 4− 2

exp(βΦ+ 2−

√β2Φ2 + 4

2

),

and is attained at the point(γΦ(β), β

)where

γΦ(β) :=βΦ− 2 +

√4 + β2Φ2

2Φ.

In addition, if we define

ΠΦ (γ, δ) :=

8

γ+

9

δ+ 9δ

e(δ−γ)Φ

γ(δ − γ),

then the infimum of ΠΦ is given by MΦ

(β) := inf(γ,δ)∈CβΠΦ

(γ, δ) = ΠΦ (γ

Φ (β), β), where γΦ

(β) is the

unique solution in (γΦ (β), β) of the equation with unknown x

8(β − x)2 − 9βe(β−x)Φ(Φx2 − (βΦ− 2)x− β

)= 0.

Moreover, it holds

limβ→∞

MΦ(β) = limβ→∞

MΦ (β) = 9eΦ.

Theorem II.14. Let (G, T, ĎX, ξ, C, f) be standard data under β. If MΦ(β) < 12 (resp. MΦ

(β) < 12 ),

then there exists a unique quadruple (Y, Z, U,N) which satisfies (II.1) and with ‖(Y, Z, U,N)‖β < ∞(resp. ‖(Y, Z, U,N)‖,β < ∞).


Remark II.15. Using the results of Lemma II.13 and Theorem II.14, it is immediate that as soon as

Φ <1

18e, (II.19)

then there always exists a unique solution of the BSDE for β large enough.

Moreover, let us now argue why the above condition rules out the counterexamples of Section II.2from our setting. The generator f(t, y, u) = 1

p (y + g(u)) needs to be Lipschitz so that it fits in our

framework, and to satisfy (II.16). Let us further assume that the function g is also Lipschitz, say withassociated constant Lg. Then, using Young’s Inequality, we can obtain

|f(t, y, u)− f(t, y′, u′)|2 ≤ 1 + ε

p2|y − y′|2 +

1

p2

(1 +

(Lg)2

ε

)|u− u′|2, for every ε > 0.

Therefore, we have that α2r = max

√1 + ε/p,

(1 + (Lg)2

ε

)/p2

, and it holds

α2r ∆Cr = max

√1 + ε,

(1 +

(Lg)2

ε

)/p≥√

1 + ε >1

18efor every ε > 0.

II.5. A priori estimates

The method of proof we will use follows and extends the one of El Karoui and Huang [29]. In [29],as well as in Pardoux and Peng [55], the result is obtained using fixed-point arguments and the so-calleda priori estimates. However, we would like to underline that the proof of such estimates in our case issignificantly harder, due to the fact that the process C is not necessarily continuous.

The following result can be seen as the a priori estimates for a BSDE whose generator does notdepend on the solution. In order to keep notation as simple as possible, as well as to make clearer thelink with the data of the problem we consider, we will reuse part of the notation of (F1)–(F5), namelyξ, T, f, C, α and A, only for the next two lemmata.

Lemma II.16. Let y be an n−dimensional G−semimartingale of the form

yt = ξ +

∫(t,T ]

fs dCs −∫ T

t

dηs, (II.20)

where T is a G−stopping time, ξ ∈ L2(GT ;Rn), f is an n−dimensional optional process, C ∈ V+pred(G;R)

and η ∈ H2(G;Rn).Let A := α2 · C for some predictable process α. Assume that there exists Φ > 0 such that property

(II.18) holds for A. Suppose there exists β ∈ R+ such that

E[eβAT ‖ξ‖22

]+ E

[∫(0,T ]

eβAt‖ft‖22α2t

dCt

]<∞.

Then we have for any (γ, δ) ∈ (0, β]2, with γ 6= δ,

‖αy‖2H2δ≤ 2eδΦ

δ‖ξ‖2L2

δ+ 2Λγ,δ,Φ

∥∥∥∥fα∥∥∥∥2

H2γ∨δ

, ‖y‖2S2δ≤ 8‖ξ‖2L2

δ+

8

γ

∥∥∥∥fα∥∥∥∥2

H2γ∨δ

,

‖η‖2H2δ≤ 9

(1 + eδΦ

)‖ξ‖2L2

δ+ 9

(1

γ ∨ δ+ δΛγ,δ,Φ

)∥∥∥∥fα∥∥∥∥2

H2γ∨δ

,

where we have defined

Λγ,δ,Φ :=1 ∨ e(δ−γ)Φ

γ|δ − γ|.

As a consequence, we have

‖αy‖2H2δ

+ ‖η‖2H2δ≤ Πδ,Φ‖ξ‖2L2

δ+ ΠΦ(γ, δ)

∥∥∥∥fα∥∥∥∥2

H2γ∨δ

, (II.21)

‖y‖2S2δ

+ ‖η‖2H2δ≤ Πδ,Φ

? ‖ξ‖2L2δ

+ ΠΦ? (γ, δ)

∥∥∥∥fα∥∥∥∥2

H2γ∨δ

, (II.22)

II.5. A PRIORI ESTIMATES 61

where

Πδ,Φ := 9 + (9 +2

δ)eδΦ and Πδ,Φ

? := 17 + 9eδΦ.

Proof. Recall the identity

yt = ξ +

∫(t,T ]

fs dCs −∫ T

t

dηs = E

[ξ +

∫(t,T ]

fs dCs

∣∣∣∣∣Gt]

(II.23)

and introduce the anticipating function

Ft :=

∫(t,T ]

fs dCs. (II.24)

For γ ∈ R+, we have by the Cauchy–Schwarz inequality,

‖Ft‖22 ≤∫

(t,T ]

e−γAs dAs

∫(t,T ]

eγAs‖fs‖22α2s

dCs ≤∫

(At,AT ]

e−γALs ds

∫(t,T ]

eγAs‖fs‖22α2s

dCs

≤∫

(At,AT ]

e−γs ds

∫(t,T ]

eγAs‖fs‖22α2s

dCs ≤1

γe−γAt

∫(t,T ]

eγAs‖fs‖22α2s

dCs, (II.25)

where for the third inequality we used Lemma I.34.(vii). For t = 0, since we assumed that

E

[∫(0,T ]

eβAt‖ft‖22α2t

dCt

]<∞,

we have that E[|F0|2

]<∞ is true for 0 < γ < β. For δ ∈ R+ and by integrating (II.25) with respect to

eδAt dAt it follows∫(0,T ]

eδAt‖Ft‖22 dAt(II.25)

≤ 1

γ

∫(0,T ]

e(δ−γ)At

∫(t,T ]

eγAs‖fs‖22α2s

dCs dAt

=1

γ

∫(0,T ]

eγAs‖fs‖22α2s

∫(0,s)

e(δ−γ)At dAt dCs ≤1

γ

∫(0,T ]

eγAs‖fs‖22α2s

∫(0,s]

e(δ−γ)At dAt dCs, (II.26)

where we used Tonelli’s Theorem in the equality. We can now distinguish between two cases:

• For δ > γ, we apply Corollary I.36 for g(x) = e(δ−γ)x, and inequality (II.26) becomes∫(0,T ]

eδAt‖Ft‖22 dAt ≤e(δ−γ)Φ

γ

∫(0,T ]

eγAs‖fs‖22α2s

∫(A0,As]

e(δ−γ)t dtdCs

≤ e(δ−γ)Φ

γ(δ − γ)

∫(0,T ]

eδAs‖fs‖22α2s

dCs, (II.27)

which is integrable if δ ≤ β.

• For δ < γ, inequality (II.26) can be rewritten as follows∫(0,T ]

eδAt‖Ft‖22 dAt ≤ 1

γ

∫(0,T ]

eγAs‖fs‖22α2s

∫(A0,As]

e(δ−γ)ALt dtdCs

Lem. I.34.(vii)

≤ 1

γ

∫(0,T ]

eγAs‖fs‖22α2s

∫(A0,As]

e(δ−γ)t dtdCs

≤ 1

γ|δ − γ|

∫(0,T ]

eγAs‖fs‖22α2s

(e(δ−γ)A0 − e(δ−γ)As

)dCs

≤ 1

γ|δ − γ|

∫(0,T ]

eγAs‖fs‖22α2s

dCs, (II.28)

which is integrable if γ ≤ β.

To sum up, for γ, δ ∈ (0, β], γ 6= δ, we have

E

[∫(0,T ]

eδAt‖Ft‖22 dAt

]≤ Λγ,δ,Φ

∥∥∥∥fα∥∥∥∥2

H2γ∨δ

. (II.29)


For the estimate of ‖αy‖H2δ

we first use the fact that

‖αy‖2H2δ

= E

[∫(0,T ]

eδAt‖yt‖22 dAt

]≤ 2E

[∫(0,T ]

E[eδAt‖ξ‖22 + eδAt‖Ft‖22

∣∣Gt]dAt

]

= 2E

[∫(0,∞)

E[eδAt‖ξ‖22 + eδAt‖Ft‖22

∣∣Gt] dATt

]Lem. I.23

= 2E

[∫(0,∞)

eδAt‖ξ‖22 + eδAt‖Ft‖22 dATt

]= 2E

[∫(0,T ]

eδAt‖ξ‖22 + eδAt‖Ft‖22 dAt

]Cor. I.36≤

(II.29)

2eδΦ

δ‖ξ‖2L2

δ+ 2Λγ,δ,Φ

∥∥∥fα

∥∥∥2

H2γ∨δ

. (II.30)

In the second equality we have used that the processes ‖ξ‖221Ω×[0,∞](·) and ‖F·‖22 are uniformly integrable,

hence their optional projections are well defined. Indeed, using (II.25) and remembering that E[‖F0‖22] <∞, we can conclude the uniform integrability of ‖F·‖22. Then, by Theorem I.22 it holds that

ΠGo

(eδA·‖ξ‖22 + eδA·‖F·‖22

)t

= eδAtΠGo

(‖ξ‖221Ω×[0,∞](·)

)t

+ eδAtΠGo

(‖F·‖22

)t

= eδAtE[‖ξ‖22

∣∣Gt]+ eδAtE[‖Ft‖22

∣∣Gt] = E[eδAt‖ξ‖22 + eδAt‖Ft‖22

∣∣Gt] ,which justifies the use of Lemma I.23.

For the estimate of ‖y‖S2δ

we have

‖y‖S2δ

= E[

sup0≤t≤T

(eδ2At‖yt‖2

)2]≤ 2E

[sup

0≤t≤TE[√

eδAt‖ξ‖22 + eδAt‖Ft‖22

∣∣∣∣Gt]2]

≤ 2E

sup0≤t≤T

E

[√eδAT ‖ξ‖22 +

1

γe(δ−γ)At

∫(t,T ]

eγAs‖fs‖22α2s

dCt

∣∣∣∣∣Gt]2

≤ 2E

sup0≤t≤T

E

[√eδAT ‖ξ‖22 +

1

γ

∫(0,T ]

e(γ+(δ−γ)+)As‖fs‖22α2s

dCs

∣∣∣∣∣Gt]2

≤ 8E

[eδAT ‖ξ‖22 +

1

γ

∫(0,T ]

e(γ∨δ)As ‖fs‖22

α2s

dCs

]≤ 8‖ξ‖2L2

δ+

8

γ

∥∥∥∥fα∥∥∥∥2

H2γ∨δ

(II.31)

for γ ∨ δ ≤ β, where in the second and third inequalities we used the inequality a+ b ≤√

2(a2 + b2) and(II.25) respectively.

What remains is to control ‖η‖H2δ. We remind once more the reader that

∫ Tt

dηs = ξ−yt+Ft, hence

E[‖ξ − yt + Ft‖2

∣∣Gt] = E

[∫(t,T ]

dTr[〈η〉s]

∣∣∣∣∣Gt]. (II.32)

In addition, we have∫(0,T ]

eδAs dTr〈η〉s =

∫(0,T ]

∫(A0,As]

δeδt dtdTr[〈η〉s] + Tr[〈η〉T ]

Lem. I.34.(vii)

≤∫

(0,T ]

∫(A0,As]

δeδALt dtdTr[〈η〉s] + Tr[〈η〉T ]

= δ

∫(0,T ]

∫(0,s]

eδAt dAtdTr[〈η〉s] + Tr[〈η〉T ]

≤ δ

∫(0,T ]

eδAt∫

(t,T ]

dTr[〈η〉s]dAt + Tr[〈η〉T ], (II.33)

so that

‖η‖H2δ≤ δE

[∫(0,T ]

eδAt∫

(t,T ]

dTr[〈η〉s] dAt

]+ E

[Tr[〈η〉T ]

]. (II.34)


For the first summand on the right-hand-side of (II.34), we have

E

[∫(0,T ]

eδAt∫

(t,T ]

dTr[〈η〉s] dAt

]Lemma I.23

= E

[∫(0,T ]

eδAtE

[∫(t,T ]

dTr[〈η〉s]

∣∣∣∣∣Gt]

dAt

](II.32)

= E

[∫(0,T ]

eδAtE[‖ξ − yt + Ft‖2

∣∣Gt] dAt

](II.32)

≤ 3E

[∫(0,T ]

eδAtE[‖ξ‖22 + ‖yt‖22 + ‖Ft‖22

∣∣Gt] dAt

](II.23)

≤ 3E

[∫(0,T ]

eδAt‖ξ‖22 dAt

]+ 3E

[∫(0,T ]

eδAt‖Ft‖22 dAt

]

+ 6E

[∫(0,T ]

eδAtE[‖ξ‖22 + ‖Ft‖22

∣∣Gt] dAt

]Lemma I.23

= 9E

[∫(0,T ]

eδAt‖ξ‖22 dAt

]+ 9E

[∫(0,T ]

eδAt‖Ft‖22 dAt

]Corollary I.36

≤(II.29)

9eδΦ

δ‖ξ‖2L2

δ+ 9Λγ,δ,Φ

∥∥∥∥fα∥∥∥∥2

H2γ∨δ

.

We now need an estimate for E[∫

(0,T ]dTr[〈η〉s]

], i.e. the second summand of (II.34), which is given by

E [Tr[〈η〉T ]] = E[‖ξ − y0 + F0‖2

]≤ 3E

[‖ξ‖22 + ‖y0‖2 + |F0|2

](II.23)

≤ 9E[‖ξ‖22

]+ 9E

[|F0|2

] (II.25)

≤ 9‖ξ‖2L2δ

+9

γ ∨ δ

∥∥∥∥fα∥∥∥∥2

H2γ∨δ

,

where we used the fact that E[‖y0‖2

]≤ 2E

[‖ξ‖22 + |F0|2

].

Then (II.34) yields

‖η‖2H2δ≤ 9

(1 + eδΦ

)‖ξ‖2L2

δ+ 9

(1

γ ∨ δ+ δΛγ,δ,Φ

)∥∥∥∥fα∥∥∥∥2

H2γ∨δ

. (II.35)

Remark II.17. An alternative framework can be provided if we define the norms in Section II.3 usinganother positive and increasing function h instead of the exponential function. In order to obtain therequired a priori estimates, we need to assume that h is sub–multiplicative7 and that it shares somecommon properties with the exponential function. However, we need to assume that the process Adefined in (F4) is P−a.s. bounded by a positive constant. We provide the detailed calculation for thecase h(x) = (1 + x)ζ , for x ∈ R+ and ζ ≥ 1 in Appendix A.1.1.

Now we are going to state in a convenient way for later use the pathwise estimates we have obtained inLemma II.16. These estimates will allow us to prove in Chapter IV that specific sequences are uniformlyintegrable. We will not need all of them in Chapter IV, however we provide all the available informationwe have.

Lemma II.18. Let ξ, T, C, α, A and Φ as in Lemma II.16. Assume that the n−dimensional G−semimartingalesy1t and y2

t can be decomposed as follows

yit = ξ +

∫(t,T ]

f is dCs −∫ T

t

dηis for i = 1, 2 (II.36)

where f1, f2 are n−dimensional G−optional processes such that

E

[∫(0,T ]

eβAt‖f it‖22α2t

dCt

]<∞,

7In the proof of Sato [64, Proposition 25.4] we can find a convenient tool for constructing sub–multiplicative functions.


for i = 1, 2 and for some β ∈ R+ and η1, η2 ∈ H2(G;Rn). Then, for γ, δ ∈ (0, β], with γ 6= δ,∫(0,T ]

eβAt‖yit‖22 dAt ≤2

βeβΦeβAT sup

t∈[0,T ]

E[∥∥ξi∥∥2

2

∣∣∣Gt]+

2

γ(β − γ)e(β−γ)Φe(β−γ)AT sup

t∈[0,T ]

E[∫

(0,T ]

eβAt

∥∥f is∥∥2

2

α2s

dCs

∣∣∣∣Gt] (II.37)

and ∫(0,T ]

eβAt‖y1t − y2

t ‖22 dAt ≤e(β−γ)Φ

γ(β − γ)e(β−γ)AT sup

t∈[0,T ]

E[∫

(0,T ]

eβAt

∥∥f1s − f2

s

∥∥2

2

α2s

dCs

∣∣∣∣Gt] (II.38)

Moreover, for the martingale parts η1, η2 ∈ H2(G;Rn) of the aforementioned decompositions, we have

supt∈[0,T ]

∥∥η1t − η1

0

∥∥2

2≤ 6 sup

t∈[0,T ]

E[∥∥ξ1

∥∥2

2+

1

β

∫(0,T ]

eβAs

∥∥f1s

∥∥2

2

α2s

dCs

∣∣∣∣Gt]+ 3

∫(0,T ]

eβAt

∥∥f1s

∥∥2

2

α2s

dCs (II.39)

and

supt∈[0,T ]

∥∥(η1t − η2

t )− (η10 − η2

0)∥∥

2

≤ 6

β

∫(0,T ]

eβAt

∥∥f1s − f2

s

∥∥2

2

α2s

dCs +3

βsupt∈[0,T ]

E[ ∫

(0,T ]

eβAs

∥∥f1s − f2

s

∥∥2

2

α2s

dCs

∣∣∣∣Gt]. (II.40)

Proof. For the following assume γ, δ ∈ (0, β] with γ 6= δ.

• We will prove Inequality (II.37) for i = 1 by following analogous to Lemma II.16 calculations. Thesole difference will be that we are going to apply the conditional form of the Cauchy–Schwartz Inequality.Moreover, by Identity (II.36), we have∥∥y1

t

∥∥2

2=

∥∥∥∥E[ξ1 +

∫(t,T ]

f1s dCs

∣∣∣∣Gt]∥∥∥∥2

2

. (II.41)

In view of these comments, we have∫(0,T ]

eβAt‖y1t ‖22 dAt

∫(0,T ]

eβAt∥∥∥∥E[ξ1 +

∫(t,T ]

f1s dCs

∣∣∣∣Gt]∥∥∥∥2

2

dAt

≤ 2

∫(0,T ]

eβAt∥∥E[ξ1

∣∣Gt]∥∥2

2dAt + 2

∫(0,T ]

eβAt∥∥∥∥E[ ∫

(t,T ]

f1s dCs

∣∣∣∣Gt]∥∥∥∥2

2

dAt

C-S Ineq.

≤ 2

∫(0,T ]

eβAt∥∥E[ξ1

∣∣Gt]∥∥2

2dAt + 2

∫(0,T ]

eβAtE[ 1

γe−γAt

∣∣∣Gt]E[∫(t,T ]

eγAt

∥∥f1s

∥∥2

2

α2s

dCs

∣∣∣∣Gt]dAt

≤ 2

∫(0,T ]

eβAtE[∥∥ξ1

∥∥2

2

∣∣∣Gt]dAt +2

γ

∫(0,T ]

e(β−γ)AtE[∫

(t,T ]

eγAt

∥∥f1s

∥∥2

2

α2s

dCs

∣∣∣∣Gt]dAt

≤ 2 supt∈[0,T ]

E[∥∥ξ1

∥∥2

2

∣∣∣Gt] ∫(0,T ]

eβAt dAt +2

γ

∫(0,T ]

e(β−γ)AtE[∫

(0,T ]

eβAt

∥∥f1s

∥∥2

2

α2s

dCs

∣∣∣∣Gt]dAt

Cor. I.36≤ 2 sup

t∈[0,T ]

E[∥∥ξ1

∥∥2

2

∣∣∣Gt] ∫(0,T ]

eβAt dAt

+2

γsupt∈[0,T ]

E[∫

(0,T ]

eβAt

∥∥f1s

∥∥2

2

α2s

dCs

∣∣∣∣Gt] ∫(0,T ]

e(β−γ)At dAt

Cor. I.36≤ 2

βeβΦeβAT sup

t∈[0,T ]

E[∥∥ξ1

∥∥2

2

∣∣∣Gt]+

2

γ(β − γ)e(β−γ)Φe(β−γ)AT sup

t∈[0,T ]

E[∫

(0,T ]

eβAt

∥∥f1s

∥∥2

2

α2s

dCs

∣∣∣∣Gt]• We will prove Inequality (II.38). We will follow analogous arguments as in the previous case, but we

are going to use instead of (II.41) the identity

‖y1t − y2

t ‖22 =

∥∥∥∥E[ ∫(t,T ]

f1s − f2

s dCs

∣∣∣∣Gt]∥∥∥∥2

2

. (II.42)


Now we have ∫(0,T ]

eβAt‖y1t − y2

t ‖22 dAt(II.42)

=

∫(0,T ]

eβAt∥∥∥∥E[ ∫

(t,T ]

f1s − f2

s dCs

∣∣∣∣Gt]∥∥∥∥2

2

dAt

C-S Ineq.

≤∫

(0,T ]

eβAtE[ 1

γe−γAt

∣∣∣Gt]E[∫(t,T ]

eγAt

∥∥f1s − f2

s

∥∥2

2

α2s

dCs

∣∣∣∣Gt]dAt

≤ 1

γ

∫(0,T ]

e(β−γ)AtE[∫

(0,T ]

eβAt

∥∥f1s − f2

s

∥∥2

2

α2s

dCs

∣∣∣∣Gt]dAt

Cor. I.36≤ e(β−γ)Φ

γ(β − γ)e(β−γ)AT sup

t∈[0,T ]

E[∫

(0,T ]

eβAt

∥∥f1s − f2

s

∥∥2

2

α2s

dCs

∣∣∣∣Gt]• Now we are going to prove (II.39) for i = 1. We use initially the analogous to Inequality (II.25) in

order to obtain ∥∥∥∥∫(0,T ]

f1s dCs

∥∥∥∥2

2

≤ 1

β

∫(0,T ]

eβAt

∥∥f1s

∥∥2

2

α2s

dCs. (II.43)

Moreover, by Identity (II.41) we obtain∥∥y1t

∥∥2

2

(II.41)

≤∥∥∥∥E[ξ1 +

∫(t,T ]

f1s dCs

∣∣∣∣Gt]∥∥∥∥2

2

≤ 2E[∥∥ξ1

∥∥2

2+

∥∥∥∥∫(t,T ]

f1s dCs

∥∥∥∥2

2

∣∣∣∣Gt](II.43)

≤ 2E[∥∥ξ1

∥∥2

2+

1

β

∫(0,T ]

eβAs

∥∥f1s

∥∥2

2

α2s

dCs

∣∣∣∣Gt]and consequently

supt∈[0,T ]

∥∥y1t

∥∥2

2≤ 2 sup

t∈[0,T ]

E[∥∥ξ1

∥∥2

2+

1

β

∫(0,T ]

eβAs

∥∥f1s

∥∥2

2

α2s

dCs

∣∣∣∣Gt]. (II.44)

Now, by Identity (II.36) we have that

supt∈[0,T ]

∥∥η1t − η1

0

∥∥2

2= supt∈[0,T ]

∥∥∥∥y1t − y1

0 +

∫(0,T ]

f1s dCs

∥∥∥∥2

2

≤ 6 supt∈[0,T ]

∥∥y1t

∥∥2

2+ 3

∥∥∥∥∫(0,T ]

f1s dCs

∥∥∥∥2

2

(II.43)

≤(II.44)

6 supt∈[0,T ]

E[∥∥ξ1

∥∥2

2+

1

β

∫(0,T ]

eβAs

∥∥f1s

∥∥2

2

α2s

dCs

∣∣∣∣Gt]+ 3

∫(0,T ]

eβAt

∥∥f1s

∥∥2

2

α2s

dCs.

• We are going to prove, now, the Inequality (II.40). We use initially the analogous to Inequality(II.25) in order to obtain∥∥∥∥∫

(0,T ]

(f1s − f2

s ) dCs

∥∥∥∥2

2

≤ 1

β

∫(0,T ]

eβAt

∥∥f1s − f2

s

∥∥2

2

α2s

dCs. (II.45)

Moreover, by Identity (II.42) we have by Conditional Cauchy-Schwartz Inequality (analogously to thesecond case)

supt∈[0,T ]

∥∥y1t − y2

t

∥∥2

2≤ 1

βsupt∈[0,T ]

E[ ∫

(0,T ]

eβAs

∥∥f1s − f2

s

∥∥2

2

α2s

dCs

∣∣∣∣Gt]. (II.46)

By Identity (II.36), we have

(η1t − η2

t )− (η10 − η2

0) = (y1t − y2

t )− (y10 − y2

0) +

∫(0,t]

(f1s − f2

s ) dCs.

Finally, we have

supt∈[0,T ]

∥∥(η1t − η2

t )− (η10 − η2

0)∥∥

2≤ supt∈[0,T ]

∥∥∥∥(y1t − y2

t )− (y10 − y2

0) +

∫(0,t]

(f1s − f2

s ) dCs

∥∥∥∥2

2

≤ 6 supt∈[0,T ]

∥∥y1t − y2

t

∥∥2

2+

3

β

∫(0,T ]

eβAs

∥∥f1s − f2

s

∥∥2

2

α2s

dCs


(II.45)

≤(II.46)

6

β

∫(0,T ]

eβAt

∥∥f1s − f2

s

∥∥2

2

α2s

dCs +3

βsupt∈[0,T ]

E[ ∫

(0,T ]

eβAs

∥∥f1s − f2

s

∥∥2

2

α2s

dCs

∣∣∣∣Gt].

Remark II.19. Viewing (II.20) as a BSDE whose generator does not depend on y and η, then thisBSDE has a solution, which can be uniquely determined by the pair (y, η). Indeed, consider the data(G, T, ξ, f, C) and the processes α and A, which all satisfy the respective assumptions of Lemma II.16 for

some β > 0. Then the semimartingale

yt = E[ξ +

∫(t,T ]

fs dCs

∣∣∣∣Gt] = E[ξ +

∫(0,T ]

fs dCs

∣∣∣∣Gt]− ∫(0,t]

fs dCs, t ∈ R+

satisfies yT = ξ and for η· := E[ξ +

∫(0,T ]

fs dCs∣∣G·]

yt − yT = E[ξ +

∫(t,T ]

fs dCs

∣∣∣∣Gt]− ξ = E[ξ +

∫(0,T ]

fs dCs

∣∣∣∣Gt]− ∫(0,t]

fs dCs − ξ

= E[ξ +

∫(0,T ]

fs dCs

∣∣∣∣Gt]+

∫(t,T ]

fs dCs −∫

(0,T ]

fs dCs − ξ = ηt +

∫(t,T ]

fs dCs − ηT .

Now, one candidate for a process X that we can choose such that (G, T,X, ξ, C, f) become standard datafor any arbitrarily chosen integrator C, is the natural pair of the zero martingale (under G), which wewill denote by 0. Hence, given the standard data (G, T, 0, ξ, C, f) the quadruple (y, Z, U, η) satisfies theBSDE

yt = ξ +

∫(t,T ]

fs dCs −∫ T

t

dηs for t ∈ J0, T K,

for any pair (Z,U). Assume now that there exists a quadruple (y, Z, U , η) which satisfies

yt = ξ +

∫(t,T ]

fs dCs −∫ T

t

dηs for t ∈ J0, T K.

Then, the pair (y − y, η − η) satisfies

y − yt = −∫ T

t

d(η − η

)s

for t ∈ J0, T K

and by Lemma II.16, for ξ = 0 and f = 0, we conclude that ‖y − y‖S2 = ‖η − η‖H2 = 0. Therefore yand y, resp. η and η, are indistinguishable, which implies our initial statement that every solution canbe uniquely determined by the pair (y, η).

In order to obtain the a priori estimates for the BSDE (II.1), we will have to consider solutions

(Y i, Zi, U i, N i), i = 1, 2, associated with the data (G, T,X, ξi, C, f i), i = 1, 2 under β, where wealso assume that f1, f2 have common r, ϑ bounds. Denote the difference between the two solutionsby (δY, δZ, δU, δN), as well as δξ := ξ1 − ξ2 and

δ2ft := (f1 − f2)(t, Y 2t , Z

2t , U

2(t, ·)), ψt := f1(t, Y 1t , Z

1t , U

1(t, ·))− f2(t, Y 2t , Z

2t , U

2(t, ·)).We have the identity

δYt = δξ +

∫(t,T ]

ψs dCs −∫ T

t

δZs dXs −∫ T

t

∫RnδU(s, ·)µ(ds,dx)−

∫ T

t

dδNs. (II.47)

For the wellposedness of this last BSDE we need the following lemma.

Lemma II.20. The processes∫ ·0

δZs dXs and

∫ ·0

∫RnδU(s, ·)µ (dt,dx)

are square-integrable martingales with finite associated ‖ · ‖β−norms.

Proof. The square-integrability is obvious. The inequalities

E [Tr[〈δZ ·X〉]] ≤ 2E[Tr[〈Z1 ·X〉]

]+ 2E

[Tr[〈Z2 ·X〉]

],

E [Tr[〈δU ? µ〉]] ≤ 2E[Tr[〈U1 ? µ〉]

]+ 2E

[Tr[〈U2 ? µ〉]

],


together with Lemma II.5 guarantee that

E

[∫(0,T ]

eβAt‖ctδZt‖22 dCt

]+ E

[∫(0,T ]

eβAt |||δUt(·)|||2t dCt

]<∞.

Therefore, by defining

Ht :=

∫ t

0

δZs dXs +

∫ t

0

∫RnδU(t, ·)µ (dt,dx) +

∫ t

0

dδNs, (II.48)

we can treat the BSDE (II.47) exactly as the BSDE (II.20), where the martingale H will play the role ofthe martingale η.

Proposition II.21 (A priori estimates for the BSDE (II.1)). Let (G, T,X, ξi, C, f i), be standard data

under β for i = 1, 2. Then ψ/α ∈ H2β

and, if MΦ(β) < 1/2, the following estimates hold

‖(αδY, δZ, δU, δN)‖2β≤ ΣΦ(β)‖δξ‖2L2

β

+ ΣΦ(β)

∥∥∥∥δ2fα∥∥∥∥2

H2β

,

‖(δY, δZ, δU, δN)‖2?,β≤ ΣΦ

? (β)‖δξ‖2L2β

+ ΣΦ? (β)

∥∥∥∥δ2fα∥∥∥∥2

H2β

,

where

ΣΦ(β) :=Πβ,Φ

1− 2MΦ(β), ΣΦ

? (β) := min

Πβ,Φ? + 2MΦ

? (β), ΣΦ(β), 8 +16

βΣΦ(β)

,

ΣΦ(β) :=2MΦ(β)

1− 2MΦ(β), ΣΦ

? (β) := min

2MΦ? (β)(1 + ΣΦ(β)),

16

β(1 + ΣΦ(β))

.

Proof. For the integrability of ψ, using the Lipschitz property (F3) of f1, f2, we get

‖ψt‖22 ≤ 2rt‖δYt‖22 + 2θct‖ctδZt‖22 + 2θdt |||δUt(·)|||2t + 2‖δ2ft‖22.

Hence by the definition of α, which implies that

r

α2≤ α2 and

θc

α2,θd

α2≤ 1, (II.49)

we get

‖ψt‖22α2t

≤ 2

(α2t ‖δYt‖22 + ‖ctδZt‖22 + |||δUt(·)|||2t +

|δ2f |2

α2

)(II.50)

≤ 2α2t ‖δYt‖22 + 2‖ctδZt‖22 + 2 |||δUt(·)|||2t

+4

α2

(‖f1(s, 0, 0,0)‖22 + rt‖Y 2

t ‖22 + θct‖ctZ2t ‖22 + θdt |||δUt(·)|||

2t

)+

4

α2

(‖f2(s, 0, 0,0)‖22 + rt‖δYt‖22 + θct‖ctδZ2

t ‖22 + θdt |||δUt(·)|||2t

)≤ 6(α2t ‖δYt‖22 + ‖ctδZt‖22 + |||δUt(·)|||2t

)+

4

α2

(‖f1(s, 0, 0,0)‖22 + ‖f2(s, 0, 0,0)‖22

),

where, having used once more that (II.49) it follows that ψα ∈ H2

β. Next, for the ‖ · ‖β−norm, we have

‖(δY, δZ, δU, δN)‖2β = ‖αδY ‖2H2β

+ ‖δZ‖2H2,β

+ ‖δU‖2H2,\

β

+ ‖δN‖2H2,⊥β

(II.48)= ‖αδY ‖2H2

β

+ ‖H‖2H2β

(II.21)

≤ Πβ,Φ‖δξ‖2L2β

+MΦ(β)∥∥ψα

∥∥2

H2β


+ 2MΦ(β)

(‖αδY ‖2H2

β

+ ‖δZ‖2H2,β

+ ‖δU‖2H2,\

β

)+ 2MΦ(β)

∥∥∥∥δ2fα∥∥∥∥2

H2β


+ 2MΦ(β)

(‖αδY ‖2H2

β

+ ‖H‖2H2β

)+ 2MΦ(β)

∥∥∥∥δ2fα∥∥∥∥2

H2β

.


Therefore, this implies

‖(αδY, δZ, δU, δN)‖2β≤ ΣΦ(β)‖δξ‖2L2

β

+ ΣΦ(β)

∥∥∥∥δ2fα∥∥∥∥2

H2β

. (II.51)

We can obtain a priori estimates for the ‖ · ‖?,β−norm by arguing in two different ways:

• The identity (II.47) gives

‖(δY, δZ, δU, δN)‖2?,β = ‖δY ‖2S2β

+ ‖δZ‖2H2,β

+ ‖δU‖2H2,\

β

+ ‖δN‖2H2,⊥β

(II.48)= ‖δY ‖2S2

β

+ ‖H‖2H2β

(II.22)

≤ Πβ,Φ? ‖δξ‖2L2

β

+MΦ? (β)

∥∥ψα

∥∥2

H2β

(II.50)


β

+ 2MΦ? (β)‖αδY ‖2H2

β

+ 2MΦ? (β)‖H‖2H2

β

+ 2MΦ? (β)

∥∥∥∥δ2fα∥∥∥∥2

H2β

(II.51)


β

+ 2MΦ? (β)

∥∥∥∥δ2fα∥∥∥∥2

H2β

+ 2MΦ? (β)

ΣΦ(β)‖δξ‖2L2β

+ ΣΦ(β)

∥∥∥∥δ2fα∥∥∥∥2

H2β

=(

Πβ,Φ? + 2MΦ

? (β)ΣΦ(β))‖δξ‖2L2

β

+ 2MΦ? (β)

(1 + ΣΦ(β)

)∥∥∥∥δ2fα∥∥∥∥2

H2β

.

• The identity (II.23) gives

‖δY ‖2S2β

= E[

sup0≤t≤T

(eβ2At‖δYt‖2

)2]

(II.23)

≤ E

[sup

0≤t≤TE[eβ2At‖δξ‖2 + e

β2At

∥∥∥∥∫(t,T ]

ψs dCs

∥∥∥∥2

∣∣∣∣Gt]2]

(II.25)

≤ 2E

[sup

0≤t≤TE[√

eβAt‖δξ‖22 +1

β

∫(t,T ]

eβAs‖ψs‖22α2s

dCs

∣∣∣∣Gt]2]

≤ 2E

[sup

0≤t≤TE[√

eβAT ‖δξ‖22 +1

β

∫(0,T ]


dCs

∣∣∣∣Gt]2]

≤ 8E

[eβAT ‖δξ‖22 +

1

β

∫(0,T ]


dCs

](II.50)

≤ 8‖δξ‖2L2β

+16

β

∥∥∥∥δ2fα∥∥∥∥2

H2β

+ ‖αδY ‖2H2β

+ ‖δZ‖2H2,β

+ ‖δU‖2H2,\

β

, (II.52)

where, in the second and fifth inequality we used the inequality a+b ≤√

2(a2 + b2) and Doob’s inequalityrespectively. Then we can derive the required estimate

‖(δY, δZ, δU, δN)‖2?,β = ‖δY ‖2S2β

+ ‖δZ‖2H2,β

+ ‖δU‖2H2,\

β

+ ‖δN‖2H2,⊥β

(II.48)= ‖δY ‖2S2

β

+ ‖H‖2H2β

(II.48)

≤(II.52)

8‖δξ‖2L2β

+16

β

∥∥∥∥δ2fα∥∥∥∥2

H2β

+16

β‖αδY ‖2H2

β

+16

β‖H‖2H2

β

(II.51)

≤(

8 +16

βΣΦ(β)

)‖δξ‖2L2

β

+16

β

(1 + ΣΦ(β)

)∥∥∥∥δ2fα∥∥∥∥2

H2β

.

II.6. Proof of the theorem

We will use now the previous estimates to obtain the existence of a unique solution using a fixedpoint argument. The reader should keep in mind that the fixed point argument boils the existence anduniqueness of the solution of the BSDE (II.1) down to a martingale representation problem.

II.6. PROOF OF THE THEOREM 69

Proof of Theorem II.14. Let (y, z, u, w) be such that (αy, z, u, w) ∈ H2β× H2,

β× H2,\

β× H2,⊥.

Then the process M defined by

M· := E

[ξ +

∫(0,T ]

f(s, ys, zs, u(s, ·)) dCs

∣∣∣∣∣G·]

+ w· ∈ H2(G;Rn),

and by Proposition I.165 it has a unique, up to indistinguishability, orthogonal decomposition

M· = M0 +

∫ ·0

ZsdXs +

∫ ·0

∫R`U(s, x) µ(ds,dx) + L·,

where (Z,U,L) ∈ H2, ×H2,\ ×H2,⊥. In view of the identity

MT −Mt =

∫ T

t

ZsdXs +

∫ T

t

∫R`U(s, x)µ(ds,dx) +

∫ T

t

dLs, 0 ≤ t ≤ T,

we obtain

E

[ξ +

∫(t,T ]


∣∣∣∣∣Gt]

= ξ +

∫(t,T ]

f(s, ys, zs, u(s, ·)) dCs −∫ T

t

ZsdXs −

∫ T

t


∫ T

t

dNs,

where N := L− w. Define now

Yt := E

[ξ +

∫(t,T ]


∣∣∣∣∣Gt].

In order to construct a contraction using Lemma II.16, we need to choose δ > γ. Then by Lemma II.13

we can choose γ? ∈ (0, β] such that inf(γ,δ)∈Cβ ΠΦ(γ, δ) = ΠΦ(γ?(β), β). Now we get that (αY,Z ·X +

U ? µ+N) ∈ H2β×H2

β, and due to the orthogonality of the martingales, recall Lemma I.164, we conclude

that (αY,Z, U,N) ∈ H2β×H2,

β×H2,\

β×H2,⊥

β. Hence, the operator

S :(H2β×H2,

β×H2,\

β×H2,⊥

β, ‖ · ‖β

)−→

(H2β×H2,

β×H2,\

β×H2,⊥

β, ‖ · ‖β

)that maps the processes (αy, z, u, w) to the processes (αY,Z, U,N) defined above, is indeed well-defined.

Let (αyi, zi, ui, wi) ∈ H2β×H2,

β×H2,\

β×H2,⊥

βfor i = 1, 2, with

S(αyi, zi, ui, wi

)= (αY i, Zi, U i, N i), for i = 1, 2.

Denote, as usual, δy, δz, δu, δw the difference of the processes and

ψt := f(t, y1

t , z1t , u

1(t, ·))− f

(t, y2

t , z2t , u

2t (·)).

It is immediate that ψα ∈ H2

βand that

‖S(αy1, z1, u1, w1

)− S

(αy2, z2, u2, w2

)‖2β

= ‖αδY ‖2H2β

+ ‖δZ‖2H2,β

+ ‖δU‖2H2,\

β

+ ‖δN‖2H2,⊥β

δξ=0

≤Lemma II.16

MΦ(β)

∥∥∥∥ψα∥∥∥∥2

H2β

(II.50)

≤ 2MΦ(β)

(‖αδy‖2H2

β

+ ‖δz‖2H2,β

+ ‖δu‖2H2,\

β

)Lemma II.16≤ 2MΦ(β)

∥∥(αy1, z1, u1, w1)−(αy2, z2, u2, w2

)∥∥2

β.

Hence, for MΦ(β) < 1/2, we can apply Banach’s fixed point theorem to obtain the existence of a

unique fixed point (Y , Z, U,N). To obtain a solution in the desirable spaces, we substitute Y in the

triplet with Y , the corresponding cadlag version; indeed, G satisfies the usual conditions and Y is asemimartingale.

The exact same reasoning using the ‖·‖S2β−norm for Y leads to a contraction when MΦ

? (β) < 1/2.

Corollary II.22 (Picard approximation). Assume that MΦ(β) < 1/2 (resp. MΦ? (β) < 1/2) and define

a sequence (Υ(p))p∈N on H2β× H2,

β× H2,\

β×H2,⊥

β(resp. on S2

β× H2,

β× H2,\

β×H2,⊥

β) such that Υ(0) is

the zero element of the product space and Υ(p+1) is the solution of


Y(p+1)t = ξ +

∫ T

t

f(s, Y (p)s , Z(p)

s , U (p)(s, ·)) dCs

−∫ T

t

Z(p+1)s dXs −

∫ T

t

∫R`U (p+1)(s, x)µ(ds,dx)−

∫ T

t

dN (p+1)s

Then

(i) The sequence (Υ(p))p∈N converges in ‖ · ‖β (resp. in ‖ · ‖?,β), to the solution of the BSDE (II.1).

(ii) The following convergence holds(Z(p), U (p)

s , N (p))−−−−−→p→∞

(Z,U,N), in Hβ(X)×Hβ(Xd)×Hβ(X⊥).

(iii) There exists a subsequence (Υ(pj))j∈N which converges dP⊗ eβAdC − a.e.

Proof. As in the proof of Theorem II.14, we obtain, for p ≥ 1,

‖Υ(p+1) −Υ(p)‖2β≤(

2MΦ(β))p‖Υ(1)‖2

β(resp. ‖Υ(p+1) −Υ(p)‖2

?,β≤(

2MΦ? (β)

)p‖Υ(1)‖2

?,β

),

(II.53)

and consequently, since∑p∈N ‖Υ(p+1)−Υ(p)‖2

β<∞ (resp.

∑p∈N ‖Υ(p+1)−Υ(p)‖2

?,β<∞), the sequence

(Υ(p))p∈N is Cauchy under ‖ · ‖β (resp. under ‖ · ‖?,β). Denote by Υ the unique limit on the product

space. Then, it coincides with the unique fixed point for the contraction S (see the proof of TheoremII.14 above) due to the construction of (Υ(p))p∈N, which proves (i).

For (ii), the result is immediate by the Cauchy property of the sequence (Υ(p))p∈N, Ito’s isometry,the stability8 of the respective closed linear space generated by X and X\ in conjunction with theirorthogonality (recall Lemma I.164), which makes H2,⊥ to be a closed subspace; see [35, Theorem 6.16].

Finally, for (iii), by the ‖ · ‖β−convergence, we can extract a subsequence (pj)j∈N such that

‖Υ(pj+1) −Υ(pj)‖β ≤ 2−2j , for every j ∈ N. (II.54)

Define, for any ε ≥ 0, Np,ε :=

(ω, t) ∈ Ω× J0, T K, |Y (p)t (ω)− Yt(ω)| > ε

. Then we have

dP⊗ eβAdC

(lim supj→∞

Npj ,ε

)= lim

j→∞dP⊗ eβAdC

∞⋃i=j

[|Y (pi) − Y | > ε

]≤ lim

j→∞

1

ε2

∞∑i=j

E

[∫ T

0

eβAt |Y (pi)t − Yt|2dCt

]

≤ limj→∞

1

ε2

∞∑i=j

‖Y (pi) − Y ‖2β

≤ limj→∞

1

ε2

∞∑i=j

( ∞∑m=1

2m∥∥∥Y (pi+m+1) − Y (pi+m)

∥∥∥2

H2β

)

≤ limj→∞

1

ε2

∞∑m=i

( ∞∑m=1

2m∥∥∥Υ(pi+m+1) −Υ(pi+m)

∥∥∥2

β

)(II.54)

≤ limj→∞

1

ε2

∞∑i=j

( ∞∑m=1

2m2−2(i+m)

)= 0, for any ε > 0.

Hence

dP⊗ eβAdC

(lim supj→∞

Npj ,0

)≤∑n∈N

dP⊗ eβAdC

(lim supj→∞

Npj ,1/n

)= 0.

Following the same arguments, we have the almost sure convergence of Zpj , Upj , Npj to the corre-sponding processes of the ‖ · ‖β−solution of the BSDE (II.1). Moreover, using the same steps, we canobtain the analogous result for the ‖ · ‖?,β−norm.

8At this point the stability is understood as “closed under stopping the processes”. For the precise statement see Heet al. [35, Definition 6.16].

II.7. RELATED LITERATURE 71

II.7. Related literature

Let us now compare our work with the papers by Bandini [6] and Cohen and Elliott [19] who alsoconsider BSDEs in stochastically discontinuous filtrations. The setting in [19] is rather different fromours. Indeed, in our case a driving martingale X is given right from the start, and as a consequence theprocess C with respect to which the generator f is integrated is linked to the predictable bracket of X.However, the authors of [19] do not choose any X from the start, but consider instead a general martingalerepresentation theorem involving countably many orthogonal martingales, which only requires the spaceof square integrable random variables on (Ω,F ,P) to be separable to hold. Furthermore, their process Ccan, unlike our case, be chosen arbitrary (in the sense that it does not have to be related to the drivingmartingales), but with the restriction that it has to be deterministic. Moreover, it has to assign positivemeasure on every interval, see Definition 5.1 therein, hence C cannot be piecewise constant; the latterwould naturally arise from a discrete-time martingale with independent increments, which is exactly thesituation one encounters when devising numerical schemes for BSDEs. Therefore, their setting cannot beembedded into our framework, and vice versa.

On the other hand, in [6], the author considers a BSDE driven by a pure–jump martingale without anorthogonal component, which is a special case of (II.1). The martingale in this setting should actually havejumps of finite activity, hence many of the interesting models for applications in mathematical finance,such as the generalized hyperbolic, CGMY, and Meixner processes, are excluded. Such a restriction isnot present at all in our framework. Otherwise, the assumptions and the conclusion in [6] are analogousto the present work. A direct comparison is however not possible, i.e. we cannot deduce the existenceand uniqueness results in her work from our setting, since the assumptions are not exactly comparable.In particular, the integrability condition (iii) on page 3 in [6] is not compatible with (F5).

Let us also compare our result with the literature on BSDEs with random terminal time. Royer [62],for instance, considers a BSDE driven by Brownian motion, where the terminal time is a [0,∞)−valuedstopping time. Hence, her setting can be embedded in ours, by assuming the absence of jumps and ofthe orthogonal component, and further requiring that C is a continuous process. She shows existenceand uniqueness of a solution under the assumptions that the generator is uniformly Lipschitz in z andcontinuous in y, and the terminal condition is bounded. Moreover, she requires that either the generatoris strictly monotone in y and f(t, 0, 0) is bounded (for all t) or that the generator is monotone in yand f(t, 0, 0) = 0 (for all t). These conditions are not directly covered by our Assumptions (F1)–(F5),however if we consider her conditions and assume in addition that the generator is Lipschitz in y, then wecan recover the existence and uniqueness result from our main theorem. Let us point out that BSDEs withconstant terminal time are related to semi–linear parabolic PDEs, while BSDEs with random terminaltime are associated to semi–linear elliptic PDEs.

We would also like to comment briefly on the choice of the norms we consider here. They are mostlyinspired from the ones defined in the seminal work of El Karoui and Huang [29], and are equivalent tothe usual norms found in the literature when the process A and time T are both bounded. Bandini [6]uses different spaces, where the norm is defined using the Doleans–Dade stochastic exponential insteadof the natural exponential. In our setting where A is allowed to be unbounded, we can only say that ournorm dominates hers. This means that we require stronger integrability conditions, but as a result wewill also obtain a solution of the BSDE with stronger integrability properties. In any case, our methodcould be adapted to this choice of the norm, albeit with modified computations in our estimates. Weremind the reader Remark II.17 for a short discussion about the definition of the norms.

Let us conclude this section by commenting on the condition (II.19). We start with the observationthat the analysis of the counterexample of Confortola et al. [22] made in Section II.2 does not allow for ageneral statement of wellposedness of the BSDE when Φ ≥ 1. In this light, the result of Cohen and Elliott[19, Theorem 6.1], which implies that the condition Φ < 1 ensures the wellposedness of the BSDE, liesin the optimal range for Φ. Analogously in the case of Bandini [6], once her results are translated usingthe Lipschitz assumption in (F3), Φ < 1√

2also ensures the wellposedness of the BSDE. On the contrary,

condition (II.19) which reads as Φ < 1/(18e), may seem much more restrictive. The first immediateremark we can make is that the stochasticity of the integrator C considerably deteriorates the conditionon Φ. In [19] the integrator is deterministic, while in [6] and in our case the integrator is stochastic.However, we would like to remind the reader, that, as explained above, the level of generality we areworking with is substantially higher than in these two references. We also want to emphasise the factthat our condition is clearly not the sharpest one possible, but we believe it is the sharpest that can beobtained using our method of proof. The main possibilities for improvement are, in our view, twofold:


• First of all, in specific situations (e.g. T bounded, f Lipschitz, less general driving processes, . . . ) onecould most probably improve the a priori estimates of Lemma II.16 by refining several of the inequalities.• Second, as highlighted in Remark II.17, we actually have a degree of freedom in choosing the norms

we are interested in. In this paper, we used exponentials, while Bandini [6] used stochastic exponentials,but other choices, leading to potentially better estimates, could also be considered.

We leave this interesting problem of finding the optimal Φ open for future research.

CHAPTER III

Stability of Martingale Representations

This chapter deals with the stability of martingale representations. In Section I.7 we have discussedabout martingale representations, therefore let us describe now roughly how do we understand their sta-bility property. For simplicity we assume that we are working on a probability space endowed with a singlefiltration F. Given, now, two sequences (Xk)k∈N and (Y k)k∈N consisting both of square-integrable mar-

tingales, we obtain for every k ∈ N the orthogonal decompositions of Y k with respect to (F, Xk,c, µXk,d

),say

Y k = Zk ·Xk,c + Uk ? µ(Xk,d,F) +Nk for k ∈ N. (III.1)

Assuming that the sequences (Xk)k∈N and (Y k)k∈N are δJ1−convergent, the martingale representations

(III.1) are stable if the sequence consisting of the orthogonal martingales (Nk)k∈N is convergent as well

as the sum of the stochastic integrals with respect to the natural pair of Xk converges to the sum ofthe stochastic integrals with respect to natural pair of X∞. Schematically, given that the convergenceindicated with the solid arrows hold, then also the convergence indicated with the dashed arrows hold.

Xk Y k = Zk ·Xk,c + Uk ? µ(Xk,F) + Nky y 99K

99K

X∞ Y∞ = Z∞ ·X∞,c + U∞ ? µ(X∞,F) + N∞

Theorem III.3 demonstrates that under the suitable framework, which is presented in Section III.1,not only the martingale representations are stable, but also their associated optional as well as predictablequadratic variation processes. The former stability is a well-known result in the literature and it is aconsequence of the fact that (Xk)k∈N, (Y k)k∈N are P-UT. However, the latter stability consists a newresult. We need to underline of course that it was Memin’s Theorems, see Theorems I.159 and I.160 or[53, Theorem 11, Corollary 12], that provided us with the stability of the canonical decompositions ofspecial semimartingales.1

This result, although interesting on its own, is also crucial for obtaining the stability of BSDEs.To be more precise, it will be the enhanced version of Theorem III.3, namely Theorem III.50, whichwe will deploy in order to obtain finally the stability of BSDEs. Commenting briefly on the connectionbetween the martingale-representation-stability and of the BSDE-stability, the reader may recall that inthe proof of Theorem II.14 we have translated the existence and uniqueness of the solution of a BSDEinto a martingale representation problem. In Chapter IV we will translate the one stability problem tothe other mutatis mutandis.

The structure of the chapter follows. In Section III.1 we set the framework and we present the maintheorem. Then, we provide in Section III.2 the outline of the proof and in the subsequent sections we willdevelop the necessary machinery for the final step of the proof, which will be presented in Section III.7.We have devoted Section III.9 for the comparison with the existing literature.

III.1. Framework and statement of the main theorem

The probability space (Ω,G,P) will be fixed throughout the chapter. The framework that we aregoing to set will be also valid for the whole chapter except for Section III.8. Let us, now, fix an arbitrarysequence of cadlag R`−valued processes (Xk)k∈N for which we assume

supk∈N

E[ ∫

(0,∞)×R`‖x‖22 µX

k

(ds,dx)

]<∞. (III.2)

1Memin mentions in [53] that this result is not completely new, since Aldous [3] has also presented an analogous one by

means of the so-called prediction process. However, the author could not access any copy of [3] in order to provide furtherdetails on this matter.

73

74 III. STABILITY OF MARTINGALE REPRESENTATIONS

Then we fix an arbitrary sequence of filtrations (Gk)k∈N with Gk := (Gkt )t≥0, for every k ∈ N, on the

probability space (Ω,G,P) and an arbitrary sequence of real-valued random variables (ξk)k∈N.

We make the following additional assumptions:

(M1) The filtration G∞ is quasi-left-continuous and the process X∞ is G∞−quasi-left-continuous.

(M2) The process Xk ∈ H2(Gk;R`) for every k ∈ N. Moreover, Xk∞

L2(G;R`)−−−−−−−→X∞∞ .(M3) The martingale X∞ possesses the G∞−predictable representation property.

(M4) The filtrations converge weakly, i.e. Gk w−−→ G∞.

(M5) The random variable ξk ∈ L2(Gk∞;R), for every k ∈ N, and ξkL2(G∞;R)−−−−−−−−→ ξ∞.

Remark III.1. In view of Proposition I.158, conditions (M2) and (M4) imply that

(Xk,Gk)ext−−−→ (X∞,G∞).

Remark III.2. In (M5), we have imposed an additional measurability assumption for the sequence ofrandom variables (ξk)k∈N, since we require that ξk is Gk∞−measurable for any k ∈ N, instead of just beingG−measurable. We could spare that additional assumption at the cost of a stronger hypothesis in (M4),namely that the weak convergence of the σ−algebrae

Gk∞w−−→ G∞∞ ,

holds in addition. To sum up, in the statement of Theorem III.3 the pair (M4) and (M5) can besubstituted by the following

(M3′) The filtrations converge weakly as well as the final σ−algebrae, that is

Gk w−−→ G∞ and Gk∞w−−→ G∞∞ .

(M4′) The sequence (ξk)k∈N ⊂ L2(G∞;R) and satisfies ξkL2(G∞;R)−−−−−−−−→ ξ∞.

Before we state the main theorem of the chapter, it would be convenient for the reader to recallNotation I.8.

Theorem III.3. Let conditions (M1) – (M5) hold and define the Gk−martingales Y k := ΠGko (ξk) =

E[ξk|Gk· ], for k ∈ N. Assume that the orthogonal decomposition of Y k with respect to(Gk, Xk,c, µX

k,d)2is given by

Y k = Y k0 + Zk ·Xk,c + Uk ? µ(Xk,d,Gk) +Nk for k ∈ N

and

Y∞ = Y∞0 + Z∞ ·X∞,c + U∞ ? µ(X∞,d,G∞) for k =∞.

Let Qk := Zk ·Xk,c + Uk ? µ(Xk,d,Gk) for k ∈ N, then the following convergence hold(Y k, Qk, Nk

) (J1(R3),L2)−−−−−−−−→(Y∞, Q∞, N∞

), (III.3)(

[Y k], [Y k, Qk], [Y k, Nk], [Y k, Xk], [Nk]) (J1(R3×R`×R),L1)−−−−−−−−−−−−−→(

[Y∞], [Y∞, Q∞], [Y∞, N∞], [Y∞, X∞], [N∞]) (III.4)

and (〈Y k〉, 〈Y k, Xk〉, 〈Qk〉, 〈Nk〉

) (J1(R×R`×R2),L1)−−−−−−−−−−−−−→(〈Y∞〉, 〈Y∞, X∞〉, 〈Q∞〉, 〈N∞〉

). (III.5)

Moreover, the three above convergence can be assumed to hold jointly in D(R9+2`). Additionally, theanalogous convergence for the terminal values of the processes appear above hold.

2Observe that we regard the random measure associated to the jumps of the purely discontinuous part Xk,d of theGk−martingale Xk, for every k ∈ N.

III.2. OUTLINE OF THE PROOF OF THE MAIN THEOREM 75

III.2. Outline of the proof of the main theorem

In this subsection we intend to ease the understanding of the technical subsections that follow. Tothis end, we present some helpful results, the strategy we are going to follow and an overview of the mainarguments we are going to use in order to prove Theorem III.3. We will reuse the introductory scheme,but now updated so that it carries the information of Conditions (M1) - (M5). However, as we havealready mentioned, the discussion we intend to do requires some preliminary results, which are going tobe also frequent referenced in the next sections. So, let us collect them in the next lemmata, which areessentially Memin’s Theorem, see Theorem I.160.

Lemma III.4. Under the framework of Theorem III.3:

(i) The convergence (Y k,Gk)ext−−−→ (Y∞,G∞) holds.

(ii) The convergence Y k0(δ|·|,P)−−−−−→ Y∞0 holds

(iii) The process 〈Y∞〉 is continuous.(iv) The convergence

(Y k, [Y k], 〈Y k〉)(J1(R×R2),L2×L1)−−−−−−−−−−−−−→ (Y∞, [Y∞], 〈Y∞〉)

holds. In particular,

Yk(J1(R×R3),L2×L1)−−−−−−−−−−−−−→ Y∞,

where Yk := (Y k, [Y k], 〈Y k〉, [Y k]− 〈Y k〉), for every k ∈ N.(v) The sequences

([Y k]∞

)k∈N,

(〈Y k〉∞

)k∈N and

(sups>0 |Y k|2

)k∈N are uniformly integrable.

(vi) The sequence (Y k)k∈N is P-UT.

Proof. (i) See Proposition I.158.(ii) It is immediate by the definition of the metric δJ1

(R).(iii) Since Y∞ is uniformly integrable, we can conclude by Theorem I.16 that it is G∞−quasi-left-

continuous. Then, we can conclude the continuity of 〈Y∞〉 by Theorem I.62.(iv) For the first convergence see Theorem I.160.(i) and combine Conditions (M1), (M4) and (M5)

with (i). We can conclude the second convergence by the continuity of 〈Y∞〉 and Corollary I.122.(v) For the sequences

([Y k]∞

)k∈N,

(〈Y k〉∞

)k∈N see Theorem I.160.(ii). The uniform integrability of

the third sequence is derived by Burkholder–Davis–Gundy Inequality, see Theorem I.101, and the de LaVallee Poussin–Meyer Criterion, see Corollary I.48.

(vi) We will apply Proposition I.153. By construction, Y k is a Gk−martingale for every k ∈ N and the

convergence Y k(J1(R),P)−−−−−−−→ Y∞ is true by (iv). It is only left to prove that supk∈N E

[sup0≤s≤t |Y ks |2

]<∞,

for every t ∈ R+. However, we have for every t ∈ R+

supk∈N

E[

sup0≤s≤t

|Y ks |2]≤ sup

k∈NE[

sups≥0|Y ks |2

]Theorem I.57.(ii)

= supk∈N

E[|Y k∞|2

]= sup

k∈NE[[Y k]∞

] (v)< ∞.

We provide the analogous results for (Xk)k∈N. We will not describe the proof as we did in the previouslemma since the arguments are completely analogous; the only difference will be that some equalities maybe inequalities whose constants depend on `.

Lemma III.5. Under the framework of Theorem III.3:

(i) The convergence (Xk,Gk)ext−−−→ (X∞,G∞) holds.

(ii) The convergence

(Xk, [Xk], 〈Xk〉)(J1(R`×R`×`×R`×`),L2×L1×L1)−−−−−−−−−−−−−−−−−−−−−−→ (X∞, [X∞], 〈X∞〉)

holds. In particular,

X k(J1(R`×R`×3`),L2×L1)−−−−−−−−−−−−−−−−→ X∞,

where X k := (Xk, [Xk], 〈Xk〉, [Xk]− 〈Xk〉) for evey k ∈ N.(iii) The process 〈X∞〉 is continuous.(iv) The sequences

(Tr[[Xk]∞

])k∈N,

(Tr[〈Xk〉∞

])k∈N and

(sups>0 ‖Xk‖22

)k∈Nare uniformly integrable.


(v) The sequence (Xk)k∈N is P-UT.

Lemma III.6. Under the framework of Theorem III.3 the convergence(Xk, Y k, [Y k, Xk], 〈Y k, Xk〉) (J1(R

×R×R×R

),P)−−−−−−−−−−−−−−−→ (X∞, Y ∞, [Y ∞, X∞], 〈Y ∞, X∞〉) (III.6)

holds.

Proof. We will apply Theorem I.160 for the sequences (Xk,i + Y k)k∈Nand (Xk,i − Y k)k∈N

, forevery i = 1, . . . , . Let us start with the following notational abuse:

Xk + Y k := (Xk,1 + Y k, . . . , Xk, + Y k) and Xk − Y k := (Xk,1 − Y k, . . . , Xk, − Y k) for k ∈ N.

Observe now that (Xk +Y k)∞, (Xk −Y k)∞ ∈ L2(G;R) for every k ∈ N. Therefore Xk +Y k, Xk −Y k ∈H2(Gk;R) and for every k ∈ N. By Conditions (M2) and (M5) we have

(Xk + Y k)∞L2(G;R)−−−−−−→ (X∞ + Y ∞)∞ as well as (Xk − Y k)∞

L2(G;R)−−−−−−→ (X∞ − Y ∞)∞

Moreover, by Lemma III.4.(iii) we have that Y ∞ is G∞−quasi-left-continuous, which in conjunction with(M1) implies that X∞ + Y ∞ and X∞ − Y ∞ are G∞−quasi-left-continuous. Now, by Proposition I.158we obtain(

Xk + Y k,Gk) ext−−−→ (

X∞ + Y ∞,G∞) as well as (Xk − Y k,Gk) ext−−−→ (

X∞ − Y ∞,G∞).Now, we can apply Theorem I.160 in order to obtain(

Xk + Y k, [Xk + Y k], 〈Xk + Y k〉,Gk) ext−−−→ (

X∞ + Y ∞, [X∞ + Y ∞], 〈X∞ + Y ∞〉,G∞)as well as(

Xk − Y k, [Xk − Y k], 〈Xk − Y k〉,Gk) ext−−−→ (

X∞ − Y ∞, [X∞ − Y ∞], 〈X∞ − Y ∞〉,G∞).Finally, by using Corollary I.125 and the polarisation identity we derive the required convergence.

Now we can proceed to outline the proof of Theorem III.3. The first part of the statement amountsto showing the following convergence

Y k = Y k0 + Zk ·Xk,c + Uk μ(Xk,d,Gk) + Nk

Y ∞ = Y ∞0 + Z∞ ·X∞,c + U∞ μ(X∞,d,G∞) + N∞

(III.7)

However, Lemma III.4 yields directly the two left convergence. Thus, the sum on the right-hand of(III.7) converges. We will be able conclude if for every weak-limit of (Nk)k∈N, name ĎN, we can determinea filtration F := (Ft)t∈R+

, which may depend on ĎN , such that the following hold:

(Α) For every t ∈ R+ holds G∞t ⊂ Ft. Alternatively written, following Notation I.11, G∞ ⊂ F.

(Β) X∞ ∈ H2(F;R), Y ∞ ∈ H2(F,R).

(Γ) [X∞]− 〈X∞〉G∞ ∈ M(F;R).

(Δ) ĎN ∈ M(F;R) and it is sufficiently integrable.

(Ε) MμX∞,d

[ΔĎN∣∣PF]= 0 and 〈X∞,c,i, ĎN c〉F = 0 for every i = 1, . . . , .

(ΣΤ) MμX∞,d [ΔY ∞∣∣PF

]= M

μX∞,d [ΔY ∞∣∣PG∞]

and 〈X∞,c,i, Y ∞〉F = 〈X∞,c,i, Y ∞〉G∞for every i =

1, . . . , .

Let us comment briefly the implications of the above conditions. By (Γ) we obtain that X∞ is alsoF−quasi-left-continuous. We will see in Subsection III.6.2 that this property will allow us to concludethat

〈X∞,c〉F = 〈X∞,c〉G∞and ν(X

∞,d,F)∣∣∣˜PG∞ = ν(X

∞,d,G∞). (III.8)

Now we will explain the outcome of specific combinations between (Α)-(ΣΤ).• The combination of (Α) and (Β) will be useful to prove (ΣΤ), which in turn will enable us to prove

that the orthogonal decomposition of Y ∞ with respect to(F, X∞,c, μX∞,d)

is indistinguishable from the

orthogonal decomposition of Y ∞ with respect to(G∞, X∞,c, μX∞,d)

. Observe that by (M2) and (Β) wehave X∞ ∈ H2(G∞;R)∩H2(F;R). Analogously, by the construction of Y ∞ and (Β) we will also obtainthat Y ∞ ∈ H2(G∞;R)∩H2(F;R). Therefore, both of the aforementioned orthogonal decompositions areindeed well-defined. This result will be presented in Subsection III.6.2.

III.3. TO BE (ORTHOGONAL), OR NOT TO BE, HERE IS THE ANSWER 77

• The combination of (Β) and (Δ) will be useful to prove (Ε). This, in turn, will enable us to provethat ĎN is orthogonal to the space generated by the natural pair of the F−martingale X∞. This result

will be provided in Subsection III.6.1. Observe that we did not claim that ĎN ∈ H2,⊥(F, X∞,c, μX∞,d

),because ĎN is not a priori square-integrable.

Exploiting these properties we will be able to identify ĎN , i.e. the arbitrary weak-limit of (Nk)k∈N, withthe zero process by proving (finally) that [Y ∞, ĎN ] ∈ M(F;R). For this conclusion, we will make use of

• The fact that the representation of Y ∞ is not affected by the expansion of the filtration.• Condition (M3). The latter can be stated alternatively as H2,⊥(G∞, Xc, μX∞,d

;R) = 0.• ĎN is orthogonal to sufficiently many stochastic integrals with respect to the natural pair of X∞.

Intuitively, this is what we need. The orthogonal martingales should converge to an element which isorthogonal to the space generated by the natural pair of X∞ and the “information that is collected fromY ∞ through stochastic integrals with respect to the natural pair of X∞ does not increase”. This stepwill be the final one of the proof and is going to be presented in Section III.7.

Let us now provide some more information on the roadway we will follow in order to obtain (Α)-(ΣΤ). Proving the properties (Α)-(ΣΤ) under the framework (M1) - (M5) requires a series of preparatoryresults. We start be providing Proposition III.7 in Section III.3. This proposition translates the problem

of determining the predictable objects 〈X∞,c,i, L〉F, for i = 1, . . . , , and MXd

[ΔL∣∣PF]into a problem of

determining a rich enough family of F−martingales. In the previously used notation, L is a uniformlyintegrable F−martingale such that the objects are well-defined. With the aforementioned proposition wewill prove in Subsection III.6.2 that (ΣΤ) is true, while with a simplification stated as Corollary III.8 wewill prove in Subsection III.6.1 that (Ε) is valid.

Having a closer look on the problem-translation mentioned above, this amounts to ensuring theexistence of a (suitable) deterministic function3 h :

(R,B(R)

) → (R+,B(R+)) and a (suitable) family

J of subsets of R such that for every I ∈ J holds

for (Ε):[(h1I) μ

(X∞,d,G∞), ĎN], [X∞,i, ĎN ] ∈ M(F;R), for every i = 1, . . . , , (III.9)

and for (ΣΤ):

[(h1I) μ

(X∞,d,G∞), Y ∞ ]− ⟨(h1I) μ(X∞,d,G∞), Y ∞⟩G∞

∈ M(F;R)

[X∞,i, Y ∞]− 〈X∞,i, Y ∞〉G∞ ∈ M(F;R), for every i = 1, . . . , .(III.10)

In order to prove (III.9) and (III.10) we will make use of Proposition I.147. Therefore, we need to weakly

approximate X∞, Y ∞, ĎN and (h1I) μ(X∞,d,G∞) for every I ∈ J . While we can weakly approximateX∞, Y ∞ and ĎN by known sequences and some subsequence, this is not the case for the martingales

(h1I) μ(X∞,d,G∞). This is the purpose of Section III.4. At the beginning of Section III.4 we will discuss

how to choose the function h and the family J and then we will present the results which provide therequired family of convergent martingale-sequences.

Let us now briefly comment on (Α), i.e. how to construct the filtration F. As we have alreadymentioned, we are going to apply Proposition I.147. In the next two lines we follow the notation ofthis proposition. Having in our mind (M4), a simple observation reveals that we can enrich the natural

filtration of the limit process H∞ provided we can improve the convergence Θk L−−→ Θ∞ to

(Θk,E[1G∞ |Gk])L−−→ (Θ∞,E[1G∞ |G∞]), for every G∞ ∈ G∞

∞ .

In other words, we will obtain that (Θ∞,E[1G∞ |G∞]) is an FE[1G∞ |G∞] ∨ FH∞-martingale, for every

G∞ ∈ G∞∞ . Then, it will be almost straightforward to obtain that Θ∞ is martingale with respect to the

filtration F := G∞ ∨FH∞. This issue will be discussed also in Section III.6. Finally, once we have all the

above results, we will present in Section III.7 the proof of Theorem III.3.

III.3. To be (orthogonal), or not to be, here is the answer

This section constitutes the first part of the preparatory results we are going to use in Section III.7.In order to avoid as many technicalities as possible amid the flow of the preparatory results, we providein this section Proposition III.7 and Corollary III.8. It is also a good chance to provide some commentsaround these results. Generally speaking, the former result provides the martingale conditions, by whichwe can conclude if the projections (under two different filtrations) of a martingale Y on the spaces

3Recall Remark I.73 for the usage of deterministic functions as predictable functions.


generated (under the two different filtration) by the natural pair of a martingale X remain unaffected.The latter provides the martingale conditions, by which we can conclude if a martingale N is orthogonalto the space generated by the natural pair of a martingale X. The probability space is (Ω,G,P), as ithas been already mentioned, but for this section we make the following assumption:

H1,H2 are fixed filtrations such that H1 ⊂ H2.

Let us now provide some more helpful comments. Assume initially that X,Y ∈M(H1,R)∩M(H2,R),that is to say, that they are both uniformly integrable martingales with respect to both given filtrations,and they are such that [X,Y ] ∈ A(H1;R) ∩ A(H2;R). By Theorem I.64 we know that there exists an

H1−predictable process [X,Y ](p,H1) of finite variation as well as an H2−predictable process [X,Y ](p,H

2)

of finite variation such that

[X,Y ]− [X,Y ](p,H1) ∈M(H1,R) and [X,Y ]− [X,Y ](p,H

2) ∈M(H2,R).

If we assume further that [X,Y ]−[X,Y ](p,H1) ∈M(H2;R), then it is a direct consequence of Corollary I.65

that [X,Y ](p,H1) = [X,Y ](p,H

2). The Proposition III.7 provides a criterion, by which we can refine theabove. In other words, it provides the conditions under which

〈Xc, Y c〉H1

= 〈Xc, Y c〉H2

and MµXd[∆Y

∣∣PH1]= MµXd

[∆Y

∣∣PH2]. (III.11)

For this, however, we need to improve the integrability of the martingales and assume that X,Y liein H2(H1,R) as well as in H2(H2,R). The integrability of X,Y and the Kunita–Watanabe Inequality,see Theorem I.100, imply that Var

([X,Y ]

)∈ A+(H1;R) ⊂ A+(H2;R). which verifies that [X,Y ] ∈

A+(H1;R) ⊂ A+(H2;R) as well. Roughly speaking, one could now translate (III.11) as “the coefficients4

of the orthogonal decompositions of Y with respect to(H1, Xc, µX

d)and the coefficients of the orthogonal

decompositions of Y with respect to(H2, Xc, µX

d)are the same”. This, however, does not imply that

the two orthogonal decompositions are indistinguishable. It may be the case that either 〈Xc〉H1 6= 〈Xc〉H2

or ν(Xd,H2)∣∣∣PH16= ν(Xd,H1), which would lead in general in different stochastic integrals. Now, Condition

(Γ) will verify that we have also (III.8).As we have already commented, we are going to apply the next proposition for the pairX∞, Y∞, while

the role of the two filtration H1,H2 will be played by G∞,F. Moreover, we anticipate once more in order to

inform the reader that Lemma III.45 ensure that 〈X∞,c〉G∞ = 〈X∞,c〉F and ν(X∞,d,F)∣∣∣PG∞

= ν(X∞,d,G∞).

So, we will also assume that the latter holds in the next proposition.

Proposition III.7. Assume that the following hold:

(i) H1 ⊂ H2, i.e. H1t ⊂ H2

t for every t ∈ R+.(ii) X ∈M(H1;R`) ∩M(H2;R`), Y ∈M(H1;R) ∩M(H2;R) and X is H1−quasi–left–continuous.

(iii) h :(R`,B(R`)

)−−→

(R+,B(R+)

)is such that h(∆Xd

t ) > 0 whenever ∆Xdt 6= 0.

(iv) ν(Xd,H2)∣∣∣PH1

= ν(Xd,H1)

(v) [X,Y ]− 〈X,Y 〉H1 ∈M(H2,R`).(vi) I is a family of subsets of R` such that σ(I) = B(R`) and the martingale (h1A) ? µ(Xd,H1)5 is

well–defined and such that (h1A) ? µ(Xd,H1) ∈ H2(H1;R) for every A ∈ I. Moreover,[(h1A) ? µ(Xd,H1), Y

]−⟨(h1A) ? µ(Xd,H1), Y

⟩H1

∈M(H2;R), for every A ∈ I.

(vii) |∆Y |µX ∈ Aσ(H1),6

Then, we have that

〈Xc,i, Y c〉H1

= 〈Xc,i, Y c〉H2

for every i = 1, . . . ` (III.12)

and

MµXd

[∆Y |PH1

] = MµXd [∆Y |PH2

] MµXd − a.e.. (III.13)

4Although this terminology sounds naive, it is not far from the reality. One should observe the way the orthogonaldecompositions are obtained and would realise that the above quantities are determining the integrands of the stochastic

integrals taken with respect to the natural pair of a square integrable martingale.5Recall Remark I.73 for the usage of deterministic functions as predictable functions.6For the notation, recall Definition I.75. Moreover, observe that (i) and (vii) imply that also |∆Y |µX ∈ Aσ(H2).

III.3. TO BE (ORTHOGONAL), OR NOT TO BE, HERE IS THE ANSWER 79

Proof. We will prove initially (III.13). To this end, we are going to translate condition (vi) into

E[(Wh∆Y

)∗ µX

d

∞

]= E

[(WhMµXd [∆Y |PH1

])∗ µX

d

∞

]for every H2−measurable function W. (III.14)

Before we proceed, recall that we have assumed X to be H1−quasi-left-continuous. Thus, by Proposi-tion I.87 we obtain that

∆((h1A) ? µ(Xd,H1)

)s

= h(∆Xd

s

)1A(∆Xd

s ) (III.15)

By the H2−martingale property of[(h1A) ? µ(Xd,H1), Y

]− 〈(h1A) ? µ(Xd,H1), Y 〉H1

, we obtain for every

0 ≤ t < u <∞ and every C ∈ H2t , that

E[1C E

[[(h1A) ? µ(Xd,H1), Y ]u − 〈(h1A) ? µ(Xd,H1), Y 〉H

1

u

∣∣∣H2t

]]= E

[1C [(h1A) ? µ(Xd,H1), Y ]t − 1C [(h1A) ? µ(Xd,H1), Y ]t

],

equivalently

E[1C E

[[(h1A) ? µ(Xd,H1), Y ]u

∣∣H2t

]− 1C [(h1A) ? µ(Xd,H1), Y ]t

]= E

[1C E

[〈(h1A) ? µ(Xd,H1), Y 〉H

1

u

∣∣H2t

]− 1C 〈(h1A) ? µ(Xd,H1), Y 〉H

1

t

],

equivalently

E[1C [(h1A) ? µ(Xd,H1), Y ]u − 1C [(h1A) ? µ(Xd,H1), Y ]t

]= E

[1C 〈(h1A) ? µ(Xd,H1), Y 〉H

1

u − 1C [(h1A) ? µ(Xd,H1), Y ]t

].

(III.16)

By Definition I.96, since (h1A) ? µ(Xd,H1) ∈ H2,d(H1;R), we have that

[(h1A) ? µ(Xd,H1), Y ]u − [(h1A) ? µ(Xd,H1), Y ]t =∑t<s≤u

(∆((h1A) ? µ(Xd,H1)

)s∆Ys

).

Moreover, in view of by Theorem I.93 and (vii), which ensures the existence of MµXd[∆Y

∣∣PH1], we can

write (III.16) equivalently

E[1C

∑t<s≤u

(∆((h1A) ? µ(Xd,H1)

)s∆Ys

)]= E

[1C

∫(t,u]×R`

(h(x)1A(x)MµXd

[∆Y

∣∣PH1](s, x)

)ν(Xd,H1)(ds,dx)

],

equivalently by (III.15),

E[ ∫

(t,u]×R`

(1Ch(x)1A(x)∆Ys

)µX

d

(ds,dx)]

= E[ ∫

(t,u]×R`

(1Ch(x)1A(x)MµXd

[∆Y

∣∣PH1](s, x)

)ν(Xd,H1)(ds,dx)

],

equivalently

E[ ∫

(0,∞)×R`

[1C1(t,u](s)1A(x)

](h(x)∆Ys

)µX

d

(ds,dx)]

= E[ ∫

(0,∞)×R`

[1C1(t,u](s)1A(x)

](h(x)MµXd

[∆Y

∣∣PH1](s, x)

)ν(Xd,H1)(ds,dx)

].

(III.17)

Now, by monotone class theorem we can conclude that condition (III.14) holds, because

PH2

= PH2

⊗ B(R`) = σ(P ×A, where P ∈ PH2

and A ∈ B(R`))

Th. I.14.(ii)= σ

(C × (t, u]×A, where 0 ≤ t < u,C ∈ H2

t and A ∈ I),

where we have from (vi) that σ(I) = B(R`).

The next observation is that the function Ω 3 (ω, s, x) 7−→ h(x) ∈ R is H2−predictable, since it is

deterministic and continuous. Moreover, the function h(x) is positive MµXd− a.e.; recall that µX

d

has

been defined using the random set [∆Xd 6= 0]. Therefore, we obtain

E[(U∆Y ) ∗ µX

d

∞

]= E

[(UM

µXd[∆Y

∣∣PH1])∗ µX

d

∞

], for every H2−measurable function U, (III.18)


by substituting in (III.14) the H2−predictable function W (ω, s, x) with the H2−predictable functionU(ω,s,x)h(x) , where U is an arbitrary H2−predictable function.

On the other hand, condition (vii) ensures the well-posedness of MµXd[∆Y

∣∣PH2]. Indeed, there

exists a positive H1−predictable function V such that E[(V∆Y ) ∗ µXd∞

]< ∞. But V , by (i), is also

PH2−measurable, which verifies our claim. By definition now of MµXd[∆Y

∣∣PH2]holds

E[(U∆Y ) ∗ µX

d

∞

]= E

[(UMµXd

[∆Y

∣∣PH2])∗ µX

d

∞

], for every H2−measurable function U. (III.19)

Now, recalling that H1 ⊂ H2, which implies that PH1 ⊂ PH2

, it is only left to observe thatMµXd[∆Y

∣∣PH1],

MµXd[∆Y

∣∣PHr] are both H2−predictable functions with

E[(UMµXd

[∆Y

∣∣PH1])∗ µX

d

∞

](III.18)

=(III.19)

E[(UMµXd

[∆Y

∣∣PH2])∗ µX

d

∞

]for every H2−measurable function U.

This implies that

MµXd[∆Y

∣∣PH1]= MµXd

[∆Y

∣∣PH2]MµXd − almost everywhere. (III.20)

We proceed now to show the validity of (III.12). By condition (ii) we have that [X,Y ]i ∈ A(H2;R) for

every i = 1, . . . , `, hence the process 〈X,Y 〉H2

is well-defined and such that [X,Y ]−〈X,Y 〉H2 ∈M(H2,R`).But then

〈X,Y 〉H2

− 〈X,Y 〉H1

= [X,Y ]− 〈X,Y 〉H1

−([X,Y ]− 〈X,Y 〉H

2)∈M(H2,R`) ∩ Vpred(H2;R`).

By Corollary I.65 it has to be indistinguishable from the zero process, i.e.

〈X,Y 〉H2

= 〈X,Y 〉H1

(III.21)

By Theorem I.93, we obtain for every i = 1, . . . , `

〈Xd,i, Y 〉H1 (III.20)

= 〈πi ? µ(Xd,H1), Y 〉H1

= (πiMµXd[∆Y |PH1

]) ∗ ν(Xd,H1)

(III.20)=

(iv)(πiMµXd

[∆Y |PH2

]) ∗ ν(Xd,H2) = 〈πi ? µ(Xd,H2), Y 〉H2

= 〈Xd,i, Y 〉H2

. (III.22)

The combination of (III.21) and (III.22) yields

〈Xc,i, Y c〉H1

= 〈Xi, Y 〉H1

− 〈Xd,i, Y d〉H1 (III.21)

=(III.22)

〈Xi, Y 〉H2

− 〈Xd,i, Y d〉H2

= 〈Xc,i, Y c〉H1

,

for every i = 1, . . . , `.

Now we can provide the following corollary, which can be proven by using completely analogous (butsimpler) arguments to the previous proposition.

Corollary III.8. Assume the framework of Proposition III.7 where (ii), (v) and (vi) are substituted bythe following

(ii′) X ∈ H2(H1;R`) ∩H2(H2;H2) and Y ∈M(F;R).(v′) [X,Y ]i ∈ A(H1;R) for every i = 1, . . . , ` and [X,Y ] ∈M(H2,R`).(vi′) I is a family of subsets of R` such that σ(I) = B(R`), the martingale (h1A)?µ(Xd,H1) is well–defined

and such that (h1A) ? µ(Xd,H1) ∈ H2(H1;R) for every A ∈ I. Moreover,[(h1A) ? µ(Xd,H1), Y

]∈M(H2;R), for every A ∈ I.

Then, we have that

〈Xc,i, Y c〉 = 0 for every i = 1, . . . ` and MµXd [∆Y |PH1

] = 0 MµXd − a.e..

Proof. The result in immediate by analogous arguments to Proposition III.7.

III.4. CONSTRUCTING A SUFFICIENTLY RICH FAMILY OF CONVERGENT MARTINGALE-SEQUENCES 81

III.4. Constructing a sufficiently rich family of convergent martingale-sequences

This section is devoted to constructing, for a given function h :(R`,B(R`)

)−→ (R+,B(R)

), a family

of sequences of purely discontinuous martingales, namely(hkI ? µ

(Xk,d,Gk))k∈N

for I ∈ J with σ(J ) = B(R`). Additionally, we will prove that for I ∈ J(Xk, hkI ? µ

(Xk,d,Gk)) (J1(R`+1),L2)−−−−−−−−−−→

(X∞, (h1I) ? µ

(X∞,d,G∞)) (III.23)

and(Xk,

[Y k, hkI ? µ

(Xk,d)]−⟨Y k, hkI ? µ

(Xk,d,⟩,Gk

) (J1(R`×R),L2×L1

−−−−−−−−−−−−−→(X∞,

[Y k, (h1I) ? µ

(Xk,d,Gk)]−⟨Y k, (h1I) ? µ

(Xk,d,⟩).

(III.24)

Recall the discussion in Section III.2 for the reason we need such a family. In particular, (III.24) isconnected with Proposition III.7.(vi). We divide this section into two subsections. In Subsection III.4.1we provide Proposition III.17, which provides the validity of (III.23), associated with its preparatoryresults. The respective results of Convergence (III.24) are presented in Subsection III.4.2, where themain result of the subsection is Proposition III.25.

Let us now comment on the choice of the function h (we need only one such function in order toapply later Proposition III.7 and Corollary III.8), of the set J and of the integrands hkI , for k ∈ N,I ∈ J . Let us start with the definition of M

µX∞,d [∆L|PF], for F a filtration such that G∞ ⊂ F, and

L ∈ M(F;R) such that ∆LµX∞,d ∈ Aσ(F). The F−predictable function M

µX∞,d [∆L|PF] is defined as

the F−predictable function such that it holds

E[(W∆L) ∗ µX

∞,d

∞]

= E[(WM

µX∞,d [∆L|PF]) ∗ µX

∞,d

∞]

for every F−predictable function W.

In the results of the previous section a positive function h has been intervened in the above products of

integrands with the property that h(∆X∞,dt ) > 0 whenever ∆X∞,dt 6= 0; we have adapted the notationof the current section. The reason is quite obvious: we have to retain the information whenever a jumphappens at a given (ω, t) ∈ [∆X∞,d 6= 0]. However, we have the flexibility to alter the jump values

∆X∞,dt (ω) in a G∞−predictable way. For example we could choose h(x) = ‖x‖1, which is positivewhenever a jump occurs for ∆X∞,d and, moreover, does not allow for any cancellations between thejumps ∆X∞,d,i, for i = 1, . . . , `. But, this is not a satisfactory choice for our purpose since we will needbetter integrability properties. The next choice would be to distinguish between “big” and “small” jumps,so we are going to examine the function

R` 3 x Rp7−→∑i=1

(|xi| ∧ 1)p ∈ R, for p ∈ [1,∞). (III.25)

where a ∧ b = mina, b for a, b ∈ R. Now, we have that∑0<s≤·

Rp(∆X∞,ds ) ≤

∑0<s≤·

‖∆X∞,ds ‖2 ∈ L1(G;R), for every p ∈ [1,∞).

This property will allow us to construct sequences of martingales with (more than) sufficient integrability.We will constraint ourselves in the case p = 2.

Now that we have chosen the function which will play the role of h, we need to approximate the

G∞−submartingale (Rp1I) ∗ µX∞,d

. We comment briefly how we obtain the approximation and thenwhy we resolve in this way. In view of Lemma III.5.(i) we can obtain the last mentioned approximationby means of Proposition I.134, which lead us also to the proper choice of J as well as of hkI , for k ∈ N,I ∈ J . If we translate Proposition I.134 to the current framework, we are allowed to choose the setsI from the set J (X∞,d), see its definition at the beginning of the next section, and the integrands hkIhave to be equal with R21I . The reader may have realised that we have obtained convergent sequences ofsubmartingales instead of martingales as we have promised. This is true. But we have done so becausethese are easier to be constructed. Indeed, using Proposition I.134 we create for every k ∈ N the process

R21I ∗µX∞,d

which is Gk−adapted, with piecewise-constant paths and with positive jumps, i.e. its path


are non-decreasing P− a.s.. Now, it is the results of Memin, more specifically Theorem I.159, which willallow us to obtain the initially required convergence

(R21I) ? µ(Xk,d,Gk) (J1,P)−−−−−→ (R21I) ? µ

(X∞,d,G∞). (III.26)

The L2−convergence will be easily obtained as a result of the choice of R2.

III.4.1. The convergence (III.23) is true. Throughout this chapter, we will consider Xk to bea Gk−martingale, for every k ∈ N. Before we proceed, let us introduce some notation that will be usedthroughout the rest of Chapter III.

Notation III.9. For a fixed i ∈ 1, . . . , `, following the notation used in Subsection I.6.1, we introducethe sets

• W (X∞,i(ω)) := u ∈ R \ 0, ∃t > 0 with ∆X∞,d,it (ω) = u.• V (X∞,i) := u ∈ R \ 0,P

([∆X∞,d,it = u, for some t > 0]

)> 0.

• I(X∞,i) := (v, w) ⊂ R \ 0, vw > 0 and v, w /∈ V (X∞,i).• J (X∞) :=

∏`i=1 Ii, where Ii ∈ I(X∞,i) ∪

R

for every i = 1, . . . , `\R`.

• For every I := I1 × · · · × I` ∈ J (X∞), we set

JI :=i ∈ 1, . . . , `, Ii 6= R 6= ∅. (III.27)

• Let k ∈ N, I := I1 × · · · × I` ∈ J (X∞) and g : Ω × R` −→ R. Then we define the R`+1−valuedprocess

Xk[g, I] :=(Xk, Xk,g,I

), with Xk,g,I := (g1I) ∗ µX

k,d

.

By Lemma I.138 we have that the set V (X∞,i) is at most countable, for every i = 1, . . . , `. Moreover,observe that an alternative representation of Xk,g,I is given by

Xk,g,I· (ω) =

∑0<t≤·

g(ω,∆Xk,dt (ω))1I(∆X

k,dt (ω)) =

∫(0,·]×R`

g(ω, x)1I(x)µXk,d

(ω; ds,dx).

Recall, moreover, that, due to (M2) and Theorem I.57.(ii), the random variable Xk∞ exists P − a.s..

Consequently, the process Xk,g,I , as well as the process Xk[g, I], are P− a.s. well–defined.

Proposition III.10. Let condition (M2) hold and fix an I ∈ J (X∞) and a function g : (Ω × R`,G ⊗B(R`)) −→ (R,B(R)). For the function g we assume that there exists ΩC ⊂ Ω with P(ΩC) = 1, and

R` 3 x 7−→ g(ω, x) ∈ R is continuous on C(ω) for every ω ∈ ΩC ,

where

C(ω) :=∏i=1

Ai(ω), with Ai(ω) :=

W (X∞,i(ω)), if i ∈ JI ,W (X∞,i(ω)) ∪ 0, if i ∈ 1, . . . , ` \ JI ,

Then, it holds

Xk[g, I](J1(R`+1),P)−−−−−−−−−−→

k→∞X∞[g, I].

Proof. Let us fix an I := I1 × · · · × I` ∈ J (X∞) and recall that the space (D(R`+1), dJ1(R`+1)) is

Polish. Therefore, it is sufficient, by Theorem I.141, to prove that for every subsequence (Xkl [g, I])l∈N,

there exists a further subsequence (Xklm [g, I])m∈N for which

Xklm [g, I]J1(R`+1)−−−−−−−−→m→∞

X∞[g, I], P− a.s.. (III.28)

Let (Xkl [g, I])l∈N be fixed hereinafter. For every i ∈ JI , since Ii ∈ I(X∞,i), there exists ΩIi ⊂ Ω suchthat

(i) P(ΩIi)

= 1,

(ii) ∆X∞,d,it (ω) /∈ ∂Ii, for every t ∈ R+ and ω ∈ ΩIi ,


where ∂A denotes the | · |−boundary of the set A.

Condition (M2) implies the convergence Xk (J1(R`),P)−−−−−−−→ X∞. Hence, for the subsequence (Xkl)l∈N,there is a further subsequence (Xklm )m∈N for which holds

XklmJ1(R`)−−−−−−→ X∞, P− a.s.. (III.29)

Let Ωsub ⊂ Ω be such that P(Ωsub) = 1 and such that the convergence (III.29) holds for every ω ∈ Ωsub.

Define ΩJIsub := Ωsub ∩(∩i∈JI ΩIi

). Then

(i′) P(ΩJIsub) = 1,

(ii′) ∆X∞,d,it (ω) /∈ ∂Ii, for every t ∈ R+, for every i ∈ JI and every ω ∈ ΩJIsub.

By the last property we can conclude that ∂Ii ∩W (X∞,i(ω)) = ∅, for every ω ∈ ΩJIsub, i ∈ JI . Therefore,

the function R` 3 xgI(ω)

7−−−→ g(ω, x)1I(x) is continuous on the set C(ω) for P−almost every ω ∈ ΩJIsub.

We can now conclude once we apply, for each ω ∈ ΩJIsub, Proposition I.134 for the sequence (Xklm (ω))m∈Nand for the function R` 3 x 7−→ g(ω, x) ∈ R, which verifies (III.28).

For the following results, recall the continuous function Rp introduced in (III.25).

Corollary III.11. Let condition (M2) hold. Then, for every I ∈ J (X∞) it holds

Xk[Rp, I](J1(R`+1),P)−−−−−−−−−→ X∞[Rp, I], for every p ∈ R+.

Proof. Let p ∈ R+. It suffices to apply the above proposition to the function Rp, which is continuous.

Let us now provide more details on the strategy of the proof for this step. Proposition III.17 be-low is the most important result of this sub–section, since it provides us with a rich enough family ofconverging martingale sequences. It is the family for which we have already commented and satisfiesconvergence (III.26). To this end, we are going to apply Theorem I.159 to the sequence (Xk,R2,I)k∈N,where I ∈ J (X∞). However, all of this requires to make sure that this sequence indeed verifies therequirements of the aforementioned theorem, for every I ∈ J (X∞). This is the subject for the remainderof this subsection. Before we proceed we provide a helpful for the calculations remark which is direct byAssumption (III.2). Then we will set some convenient notation.

Remark III.12. Let us fix initially a k ∈ N. In view of Assumption (III.2) we obtain that

E[Tr[q] ∗ µX

k

∞

]= E

[Tr[q] ∗ ν(Xk,Gk)

∞

]. (III.30)

This implies that we can associate to the process Xk a square-integrable Gk−martingale, which in (M2)we have denoted with the same symbol Xk. 7 For the calculations, this means that

∑s>0 ‖∆Xk,d‖22 ∈

A+(Gk;R). Now, letting k run through N, we have that Assumption (III.2) and the Equality (III.30)yield that

supk∈N

E[Tr[q] ∗ µX

k

∞

]= sup

k∈NE[Tr[q] ∗ ν(Xk,Gk)

∞

]<∞,

which in turn, in view of Proposition I.85, implies

supk∈N

E[Tr[[Xk,d

]∞

]]= sup

k∈NE[∑s>0

∥∥∆Xk,ds

∥∥2

2

]= sup

k∈NE[Tr[q] ∗ µX

k,d

∞

]= sup

k∈NE[∑s>0

∆(

Tr[q] ∗ µ(Xk,Gk))s

]≤ sup

k∈NE[Tr[q] ∗ µX

k

∞

]+ supk∈N

E[Tr[q] ∗ ν(Xk,Gk)

∞

]<∞. (III.31)

We end this remark with following comment. The reader may recall that from Lemma III.5.(iv) we havebetter integrability properties for the sequence

(Tr[[Xk,d

]∞

])k∈N, however the above bound is immediate

by Assumption (III.2) and we would prefer to refer to the above bounds whenever these are sufficient forour needs.

Let us, now, fix I :=∏`i=1 Ii ∈ J (X∞), hence the set JI is a fixed non–empty subset of 1, . . . , `.

Moreover, we define R` 3 x R2,A7−−−→ R2(x)1A(x), for every A ⊂ R`.7We underline again at this point that, according to our notation, the integer-valued measure µX

kassociates to the

jumps of the process Xk, while the integer valued measure µXk,d

associates to the jumps of the Gk−martingale Xk, whosepurely discontinuous part is Xk,d.


Lemma III.13. The process Xk,R2,I is a Gk−special semimartingale, for every k ∈ N. In particular, itsGk–canonical decomposition is given by

Xk,R2,I· =

∫ ·0

∫R`

R2,I(x)µ(Xk,d,Gk)(ds,dx) +

∫(0,·]×I

R2(x)ν(Xk,d,Gk)(ds,dx)

or equivalently

R2,I ∗ µXk,d

= R2,I ? µ(Xk,d,Gk) + R2,I ∗ ν(Xk,d,Gk).

Proof. Let k ∈ N. Observe that by construction the process Xk,R2,I is Gk−adapted and cadlag.The function R2,I is positive, hence the process Xk,R2,I is a Gk−submartingale of finite variation, as its

paths are P− a.s. non–decreasing. Before we proceed, we need to show the integrability of R2,I ∗ µXk,d

∞in order to make use of Proposition I.85. But, we have

E[ ∫

(0,∞)×R`R2,I(x)µX

k,d

(ds,dx)

]≤ E

[ ∫(0,∞)×R`

‖x‖2µXk,d

(ds,dx)

]= E

[T[[Xk,d

]]] (III.31)

≤ ∞, (III.32)

where the last equality holds because of Theorem I.79. Moreover, (III.32) yields also that Xk,R2,I ∈Ssp(Gk;R), by Proposition I.98.(ii). We concluded the last property because the process in increasing.Moreover, by Theorem I.79 and Condition (III.32) (which implicitly makes use of (III.12)) we obtain

E[ ∫

(0,∞)×R`R2,I(x) ν(Xk,d,Gk)(ds,dx)

]<∞, for every k ∈ N. (III.33)

Therefore, we have

Xk,R2,I· = (R21I) ∗ µX

k,d

=[(R21I) ∗ µX

k,d

− (R21I) ∗ ν(Xk,d,Gk)]

+ (R21I) ∗ ν(Xk,d,Gk)

= (R21I) ? µ(Xk,d,Gk) + (R21I) ∗ ν(Xk,d,Gk)

= R2,I ? µ(Xk,d,Gk) + R2,I ∗ ν(Xk,d,Gk),

where in the third equality we have used Proposition I.85. The finite variation part R2,I ∗ν(Xk,d,Gk) is pre-

dictable, since R2,I is deterministic and the random measure ν(Xk,d,Gk) is predictable (see Definition I.10and Theorem I.79). Hence, we can conclude also via this route that Xk,R2,I is a special semimartingale,since it admits a representation as the sum of a martingale and a predictable part of finite variation.

Lemma III.14. (i) The sequence(Xk,R2,I∞


(ii) The sequence([Xk,R2,I ]∞


Proof. (i) Using the definitions of R2 and Xk,R2,I , we get

Xk,R2,I∞ =

∫(0,∞)×R`

R2,I(x)µXk,d

(ds,dx) ≤∫

(0,∞)×R`|x|2 µX

k,d

(ds,dx) = Tr[[Xk,d]∞

]≤ Tr

[[Xk]∞

].

Hence, from Lemma III.5.(iv) and Corollary I.29.(ii) we can conclude.

(ii) By Lemma III.13, the process Xk,R2,I is a Gk−special semimartingale for every k ∈ N, whosemartingale part is purely discontinuous. Therefore, we have by Definition I.96 that

[Xk,R2,I ]∞ =∑s>0

∣∣R2

(∆Xk,d

s

)∣∣21I(∆Xk,ds ) ≤

∑s>0

∣∣R2

(∆Xk,d

s

)∣∣2 =∑s>0

(∑i=1

(∣∣∆Xk,d,is

∣∣ ∧ 1)2)2

=∑s>0

[∑i=1

(∣∣∆Xk,d,is

∣∣21(0,1)

(∣∣∆Xk,d,is

∣∣)+ 1[1,∞)

(∣∣∆Xk,d,is

∣∣))]2≤ 2`

∑s>0

[∑i=1

(∣∣∆Xk,d,is

∣∣41(0,1)

(∣∣∆Xk,d,is

∣∣)+ 1[1,∞)

(∣∣∆Xk,d,is

∣∣))]

≤ 2`∑s>0

[∑i=1

(∆Xk,d,i

s

)2]= 2`Tr

[[Xk,d]∞

]≤ 2`Tr

[[Xk]∞

].

Thus, using Lemma III.5.(iv) and Corollary I.29.(ii) again, we have the required result.


Lemma III.15. (i) The sequence(Var(R2,I ∗ ν(Xk,d,Gk))∞

)k∈N is tight in (R, | · |).

(ii) The sequence(∑

s>0

(∆(R2,I ∗ ν(Xk,d,Gk))s

)2)k∈N is uniformly integrable.

Proof. (i) We have already observed that Xk,R2,I is a Gk−submartingale for every k ∈ N and

consequently R2,I ∗ν(Xk,d,Gk) is non–decreasing for every k ∈ N; a property which is also immediate since

R2,I is a positive function and ν(Xk,d,Gk) is a (positive) measure. Therefore, it holds

Var(R2,I ∗ ν(Xk,d,Gk)

)= R2,I ∗ ν(Xk,d,Gk), for every k ∈ N.

In view of the above and due to Markov’s inequality, it suffices to prove that supk∈N E[R2,I ∗ν(Xk,d,Gk)∞ ] <

∞. Indeed, for every ε > 0 it holds for K := 1ε supk∈N E[R2,I ∗ ν(Xk,d,Gk)

∞ ] > 0 that

supk∈N

P[R2,I ∗ ν(Xk,d,Gk)

∞ > K]≤ 1

Ksupk∈N

E[R2,I ∗ ν(Xk,d,Gk)∞ ] < ε,

which yields the required tightness. Now, observe that we have

E[R2,I ∗ ν(Xk,d,Gk)∞ ]

Proposition I.85= E[R2,I ∗ µX

k,d

∞ ](III.32)< ∞. (III.34)

We have concluded using inequality (III.32), which in turn makes use of Assumption (III.2). There-

fore (III.34) yields that supk∈N E[R2,I ∗ ν(Xk,d,Gk)∞ ] < ∞. (ii) Before we proceed, the reader may recall

Remark I.80, i.e. that we use a version of ν(Xk,d,Gk) for which ν(Xk,d,Gk)(s × R` ≤ 1 identically forevery k ∈ N. In view of this property, we can apply Cauchy–Schwartz Inequality in order to obtain thefollowing upper bound(∫

R`R2,I(x) ν(Xk,d,Gk)(s × dx)

)2

≤∫R`

R22,I(x) ν(Xk,d,Gk)(s × dx)

∫R`ν(Xk,d,Gk)(s × dx)

≤∫R`

R22,I(x) ν(Xk,d,Gk)(s × dx). (III.35)

Now, we have, for every k ∈ N, that the following holds:∑s>0


)2=∑s>0

(∫R`

R2,I(x) ν(Xk,d,Gk)(s × dx))2

(III.35)

≤∑s>0

∫R`

R22,I(x) ν(Xk,d,Gk)(s × dx) ≤

∫(0,∞)×R`

R22,I(x) ν(Xk,d,Gk)(ds,dx)

=

∫(0,∞)×I

[∑i=1

(|xi| ∧ 1)2]2ν(Xk,d,Gk)(ds,dx) ≤ 2`

∫(0,∞)×I

∑i=1

(|xi|2 ∧ 1) ν(Xk,d,Gk)(ds,dx)

= 2`R2,I ∗ ν(Xk,d,Gk)∞ . (III.36)

Using Lemma III.14.(i) and the de La Vallee Poussin–Meyer Criterion, see Corollary I.48, there existsa moderate Young function Φ such that

supk∈N

E[Φ(Xk,R2,I∞

)]<∞.

Then, using that Xk,R2,I is an increasing process, hence it is equal to its supremum process, the decom-position of Lemma III.13 and applying Theorem I.55 we can conclude that

supk∈N

E[Φ(R2,I ∗ ν(Xk,d,Gk)

∞)]<∞.

By de La Vallee Poussin–Meyer Criterion, the latter condition is equivalent to the uniform integrability of

the sequence(R2,I ∗ν(Xk,d,Gk)

∞)k∈N. Then, by (III.36) and Corollary I.29.(ii) we can conclude the uniform

integrability of the required sequence.

Corollary III.16. (i) The sequence([

R2,I ? µ(Xk,d,Gk)

]∞


(ii) Consequently, it holds that R2,I ? µ(Xk,d,Gk) ∈ H2,d(Gk;R), for every k ∈ N.

(iii) The sequence(〈R2,I ? µ

(Xk,d,Gk)〉∞)k∈N is uniformly integrable.


Proof. (i) Using Definition I.96 and that R2,I ? µ(Xk,d,Gk) is a martingale of finite varation, we

have[R2,I ? µ

(Xk,d,Gk)]∞ =

∑s>0

(∆(R2,I ? µ

(Xk,d,Gk))s

)2

=∑s>0

(∆(Xk,R2,I)s −∆(R2,I ∗ ν(Xk,d,Gk))s

)2

≤ 2∑s>0

(∆(Xk,R2,I)s

)2

+ 2∑s>0


)2

= 2 [Xk,R2,I ]∞ + 2∑s>0


)2

,

where in the last equality we have used that Xk,R2,I is a semimartingale whose paths have finite variationand Definition I.96. In view now of the above inequality, Lemma III.14.(ii), Lemma III.15.(ii) and (ii),(iv)of Corollary I.29 we can conclude the required property. This shows (i).

(ii) In addition, (i) implies the integrability of[R2,I ? µ

(Xk,d,Gk)], hence from [41, Proposition I.4.50.c)]

we get that R2,I ? µ(Xk,d,Gk) ∈ H2,d(Gk,∞;R).

(iii) It is immediate by (i) and Theorem I.55.

Proposition III.17. Let conditions (M1), (M2) and (M4) hold. Then the following convergence holds:(Xk,R2,I ? µ

(Xk,d,Gk)) (J1(R`+1),L2)−−−−−−−−−−−→

(X∞,R2,I ? µ

(X∞,d,G∞)). (III.37)

Proof. As we have already pointed out on page 83, we are going to apply Theorem I.159 to thesequence (Xk,R2,I)k∈N. By Lemma III.13, this is a sequence of Gk−special semimartingales, for every

k ∈ N.

In view of (M1), which states that X∞ is G∞−quasi–left–continuous, and using Proposition I.87, we

get that the compensator ν(X∞,d,G∞) associated to µX∞,d

is an atomless random measure. Therefore, thefinite variation part of the G∞−canonical decomposition of X∞,R2,I is a continuous process. Moreover,by Theorem I.16 and (M1), which states that the filtration G∞ is quasi–left–continuous, it suffices toshow that the martingale part of X∞,R2,I is uniformly integrable. The latter holds by Corollary III.16.(ii).

Lemma III.14.(ii) yields that condition (i) of Theorem I.159 holds. Lemma III.15.(i) yields thatcondition (ii) of the aforementioned theorem also holds. Moreover, from Corollary III.11 for p = 2, weobtain the convergence (

Xk, Xk,R2,I) (J1(R`+1),P)−−−−−−−−−−→

(X∞, X∞,R2,I

). (III.38)

The last convergence in conjunction with conditions (M2) and (M4), Remark III.1 and Corollary I.125, is

equivalent to the convergence (Xk,R2,I ,Gk)ext−−−−→ (X∞,R2,I ,G∞). Therefore, condition (iii) of Theorem

I.159 is also satisfied.

Applying now Theorem I.159 to the sequence (Xk,R2,I)k∈N, and keeping in mind the decompositionfrom Lemma III.13, we obtain the convergence(

Xk,R2,I ,R2,I ? µ(Xk,d,Gk)

) (J1(R2),P)−−−−−−−−→(X∞,R2,I ,R2,I ? µ

(X∞,d,G∞)). (III.39)

Using Corollary I.125, we can combine the convergence in (III.38) and (III.39) to obtain(Xk,R2,I ? µ

(Xk,d,Gk)) (J1(R`×R),P)−−−−−−−−−−→

(X∞,R2,I ? µ

(X∞,d,G∞)).

Observe that, in view of the last three convergence and Corollary I.125, we can actually conclude theconvergence (

Xk,R2,I ? µ(Xk,d,Gk),Gk

) ext−−−−→(X∞,R2,I ? µ

(X∞,d,G∞),G∞), (III.40)

which will be useful for later reference.

The last result can be further strengthened to an L2−convergence in view of the following arguments:

Let αk =(Xk,R2,I ? µ

(Xk,d,Gk)), then by Vitali’s Convergence Theorem, see Theorem I.32, the latter is

equivalent to showing that d2J1

(αk, α∞) is uniformly integrable. Moreover, by the inequality

d2J1

(αk, α∞) ≤(dJ1(αk, 0) + dJ1(0, α∞)

)2 ≤ 2d2J1

(αk, 0) + 2d2J1

(0, α∞) ≤ 2‖αk‖2∞ + 2‖α∞‖2∞,


and Corollary I.29.(ii),(iv), it suffices to show that (‖αk‖2∞)n∈N is uniformly integrable.

By Corollary III.16.(i) we know that the sequence([R2,I ? µ

(Xk,Gk)]∞)k∈N is uniformly integrable.

Therefore, using de La Vallee Poussin–Meyer Criterion, there exists a moderate Young function φ suchthat

supk∈N

E[φ([

R2,I ? µ(Xk,Gk)

]∞

)]<∞. (III.41)

Proposition I.51 yields that the map R+ 3 xψ7−→ ψ(x) := φ( 1

2x2) is again moderate and Young. We can

apply now the Burkholder–Davis–Gundy (BDG) inequality Theorem I.101 (in conjunction with Propo-

sition I.53.(v)) to the sequence of martingales(R2,I ? µ

(Xk,Gk))k∈N using the function ψ, and we obtain

that

supk∈N

E[φ(1

2sups>0

∣∣(R2,I ? µ(Xk,Gk))s

∣∣2)] = supk∈N

E[ψ(

sups>0

∣∣(R2,I ? µ(Xk,Gk))s

∣∣)]BDG≤ Cψ sup

k∈NE[ψ([

R2,I ? µ(Xk,Gk)

] 12

∞

)]= Cψ sup

k∈NE[φ(1

2

[R2,I ? µ

(Xk,Gk)]∞

)] (III.41)< ∞. (III.42)

Hence the sequence(

sups>0 |R2,I ? µ(Xk,Gk)|2

)k∈N is uniformly integrable, again from de La Vallee

Poussin–Meyer Criterion. Moreover,(Tr[[Xk]∞

])k∈N is a uniformly integrable sequence; it was proven

in Lemma III.5.(iv). Using analogous arguments and recalling to the above inequality, we can concludethat the sequence

(sups>0 ‖Xk

s ‖22)k∈N is also uniformly integrable. Hence, the family(sups>0|Xk

s |2 + sups>0

∣∣R2,I ? µ(Xk,Gk)

∣∣2)k∈N

is uniformly integrable, which allows us to conclude.

Lemma III.18. The sequence (R2,I ? µ(Xk,d,Gk))k∈N possesses the P-UT property, consequently(

R2,I ? µ(Xk,d,Gk), [R2,I ? µ

(Xk,d,Gk)]) (J1(R2),P)−−−−−−−→

(R2,I ? µ

(X∞,d,G∞), [R2,I ? µ(X∞,d,G∞)]

)(III.43)

Proof. We will apply Proposition I.153. To this end, let us verify that the requirements are fulfilled.

Firstly recall that R2,I ? µ(X∞,d,G∞) is G−martingale, for every k ∈ N. Then, (III.42) yields that the

sequence (R2,I ? µ(X∞,d,G∞))k∈N is L2−bounded. Finally, (III.39) verifies the convergence

R2,I ? µ(Xk,d,Gk) (J1(R),P)−−−−−−−→ R2,I ? µ

(X∞,d,G∞).

III.4.2. The convergence (III.24) is true. Now that we made our choice for h, the convergence(III.24) reads(Xk,

[Y k,R2,I ? µ

(Xk,d,Gk)]−⟨Y k,R2,I ? µ

(Xk,d,Gk)⟩) (J1(R`×R),L2×L1)−−−−−−−−−−−−−→(

X∞,[Y k,R2,I ? µ


(Xk,d,Gk)⟩).

(III.44)

We comment once more that (III.44) will be used later to apply Proposition III.7. As one would expect,the strategy is very similar to the previous subsection. In other words, we will apply Theorem I.159 for

the sequences([Y k + R2,I ? µ

(Xk,d,Gk)])k∈N and

([Y k − R2,I ? µ

(Xk,d,Gk)])k∈N and then we will use the

polarisation identity to conclude.

Notation III.19. The following notation will be valid only for this subsection.

• Rk,2,I := R2,I ? µ(Xk,d,Gk), for every k ∈ N.

• Sk,+ :=[Y k + Rk,2,I

]∈ A+(Gk;R), for every k ∈ N.

• Sk,− :=[Y k − Rk,2,I

]∈ A+(Gk;R), for every k ∈ N.

The following sequence of lemmata amounts to verifying that the requirements of Theorem I.159 arefulfilled for the sequences (Sk,+)k∈N and (Sk,−)k∈N. Obviously, we will provide the proofs only for theformer sequence, since the arguments will be exactly the same for the latter.


Lemma III.20. For every k ∈ N holds Sk,+, Sk,− ∈ Ssp(Gk;R) with canonical decomposition

Sk,+ = Mk,+ + 〈Y k + Rk,2,I〉 and Sk,− = Mk,− + 〈Y k − Rk,2,I〉, (III.45)

where Mk,+,Mk,− ∈M(Gk;R), 〈Y k + Rk,2,I〉, 〈Y k − Rk,2,I〉 ∈ A+pred(Gk;R) for every k ∈ N.

Proof. By Lemma III.4.(v) and Lemma III.14.(i) we have that Y k,Rk,2,I ∈ H2(Gk;R) for every

k ∈ N. Therefore, Y k + Rk,2,I ∈ H2(Gk;R) as well, for every k ∈ N. By Proposition I.99 we have thatSk,+ ∈ A+(Gk;R), which in virtue of Proposition I.98 implies that Sk,+ ∈ Ssp(Gk;R), for every k ∈ N.

Now, again by Proposition I.99 we obtain the Gk−canonical decomposition

Sk,+ =[Y k + Rk,2,I

]= Mk,+ + 〈Y k + Rk,2,I〉,

for some uniformly integrable Gk−martingale, where the required properties hold for every k ∈ N.

Lemma III.21. The processes S∞,+, S∞,− are G∞−quasi-left-continuous.

Proof. Since Y∞,R∞,2,I ∈ H2(G∞;R) ⊂M(G∞;R), we obtain by Theorem I.16 that the R2−valued

martingale (Y∞,R∞,2,I) is G∞−quasi-left-continuous. Therefore, by Proposition I.19 there exists a se-

quence (τm)m∈N of G∞−totally inaccessible stopping times which exhausts the jumps of (Y∞,R∞,2,I).On the other hand, for every G∞−stopping time τ holds

∆S∞τ = ∆[Y∞ + R∞,2,I ]τ =(∆(Y∞ + R∞,2,I)τ

)2=(∆Y∞τ + ∆R∞,2,Iτ

)2.

Therefore, the sequence (τm)m∈N exhausts also the jumps of S∞ and now we can conclude by usingProposition I.19 again.

Lemma III.22. The sequences ([Sk,+]12∞)k∈N, ([Sk,−]

12∞)k∈N are uniformly integrable.

Proof. We will dominate the sequence of positive random variables ([Sk,+]∞)k∈N by a uniformly

integrable sequence; recall Corollary I.29.(ii). Observe that, since Sk,+ ∈ A+(Gk;R), then by Defini-

tion I.96 holds [Sk,+]∞ =∑s>0

(∆Sk,+s

)2. Therefore,

[Sk,+]12∞ =

(∑s>0

(∆Sk,+s

)2) 12

=(∑s>0

(∆(Y k + Rk,2,I)2

s

)2) 12

=(∑s>0

(∆Y ks + ∆Rk,2,I

s

)4) 12 8

≤∑s>0

(∆Y ks + ∆Rk,2,I

s

)2 ≤ 2∑s>0

(∆Y ks

)2+ 2

∑s>0

(∆Rk,2,I

s

)2 ≤ 2[Y k]∞ + 2[Rk,2,Is ].

By Lemma III.4.(v) we have that ([Y k]∞)k∈N is uniformly integrable and by Lemma III.14.(i) we recall

the uniform integrability of ([Rk,2,I ]∞)k∈N. Now, we use (i),(iv) and (ii) of Corollary I.29 to conclude thedesired uniform integrability.

Lemma III.23. The sequences (Sk,+∞ )k∈N, (Sk,−∞ )k∈N are uniformly integrable. As a particular conse-

quence, the sequences(〈Y k + Rk,2,I〉∞

)k∈N,

(〈Y k − Rk,2,I〉∞

)k∈N are uniformly integrable and tight in

R.

Proof. We start by proving that the sequence (Sk,+∞ )k∈N is uniformly integrable. To this end we

use that Sk,+ is increasing in order to dominate the terminal Sk,+∞ by elements consisting a uniformlyintegrable family. We have

0 ≤ Sk,+∞ = [Y k + Rk,2,I ]∞ = [Y k]∞ + 2[Y k,Rk,2,I ]∞ + [Rk,2,I ]∞

≤ [Y k]∞ + 2Var([Y k,Rk,2,I ]

)∞ + [Rk,2,I ]∞

≤ [Y k]∞ + 2[Y k]12∞[Rk,2,I ]

12∞ + [Rk,2,I ]∞

≤ [Y k]∞ + [Y k]∞ + [Rk,2,I ]∞ + [Rk,2,I ]∞

= 2[Y k]∞ + 2[Rk,2,I ]∞.

In the second inequality we used the Kunita–Watanabe Inequality, see Theorem I.100. In the third in-equality we used Young Inequality, see Lemma I.44.(i), applied for the pair of Young functions (quad, quad?) =

8We use ω−wise the property ‖ · ‖ϑ2≤ ‖ · ‖ϑ1

for ϑ1 ≤ ϑ2, where the norms are those associated to the classical spaces

`ϑ(N) :=x := (xk)k∈N ⊂ R, ‖x‖ϑ :=

(∑k∈N |xk|ϑ

) 1ϑ <∞

.


(quad, quad), see Example I.46. Now, by Lemma III.4.(v), Lemma III.14.(ii) and Corollary I.29 we con-clude.

Before we proceed to prove the uniform integrability of the sequence(〈Y k + Rk,2,I〉∞

)k∈N, we denote

by Φ the moderate Young function associated to the uniformly integrable family (Sk,+∞ )k∈N by the de La

Vallee Poussin–Meyer Criterion, see Corollary I.48. Then, it holds supk∈N E[Φ(Sk,+∞ )

]<∞. Now, by the

canonical decompositions (III.45) and Theorem I.55 we have that

supk∈N

E[Φ(〈Y k + Rk,2,I〉∞)

]≤ (2c

Φ)cΦ sup

k∈NE[Φ(Sk,+∞ )

]<∞.

In other words, we have that Φ and(〈Y k+Rk,2,I〉∞

)k∈N satisfy the de La Vallee Poussin–Meyer Criterion.

Therefore,(〈Y k+Rk,2,I〉∞

)k∈N is also uniformly integrable. Finally, the sequence

(〈Y k+Rk,2,I〉∞

)k∈N is

tight in R as a direct consequence of Markov’s Inequality and the L1−boundedness of(〈Y k+Rk,2,I〉∞

)k∈N;

by Theorem I.25.

Lemma III.24. The following convergence are valid

(Xk, Sk,+,Gk)ext−−−→ (X∞, S∞,+,G∞) and (Xk, Sk,−,Gk)

ext−−−→ (X∞, S∞,−,G∞)

Proof. We need to prove initially the joint convergence of (Y k,Rk,2,I)k∈N. In view of the alreadyproven convergence (we provide their associated numbering)(

Xk, Y k, [Y k, Xk], 〈Y k, Xk〉) (J1(R`×R×R`×R`),P)−−−−−−−−−−−−−−−→

(X∞, Y∞, [Y∞, X∞], 〈Y∞, X∞〉

)(III.6)

and (Xk,R2,I ? µ

(Xk,d,Gk),Gk) ext−−−−→

(X∞,R2,I ? µ

(X∞,d,G∞),G∞), (III.40)

we can conclude the convergence(Xk, Y k,Rk,2,I ,Gk

) ext−−−−→(X∞, Y∞,R∞,2,I ,G∞

).

The conclusion was drawn by Corollary I.125, because they share the sequence (Xk)k∈N. The sequences

(Y k)k∈N, (Rk,2,I)k∈N are P-UT by Lemma III.4.(vi), Lemma III.18. Then, by Remark I.150 we have that

(Y k+Rk,2,I)k∈N is also P-UT. On the other hand, by Corollary I.122, Theorem I.152 and the last remarkwe can conclude that(

Xk, Y k,Rk,2,I , Y k + Rk,2,I , Sk,+,Gk) ext−−−−→

(X∞, Y∞,R∞,2,I , Y∞ + R∞,2,I , S∞,G∞

), (III.46)

which implies the required convergence.

Proposition III.25. The following convergence is valid(Xk,

[Y k,R2,I ? µ


(Xk,d,Gk)⟩) (J1(R`×R),L2×L1)−−−−−−−−−−−−−→(

X∞,[Y k,R2,I ? µ


(Xk,d,Gk)⟩).

(III.44)

Proof. In view of Lemma III.20 - Lemma III.24 we apply Theorem I.159 for the sequences (Sk,+)k∈Nand (Sk,−)k∈N. Therefore(

Xk, [Y k + Rk,2,I ], 〈Y k + Rk,2,I〉,Gk) ext−−−−→

(X∞, [Y∞ + R∞,2,I ], 〈Y∞ + R∞,2,I〉,G∞

)and (

Xk, [Y k − Rk,2,I ], 〈Y k − Rk,2,I〉,Gk) ext−−−−→

(X∞, [Y∞ − R∞,2,I ], 〈Y∞ − R∞,2,I〉,G∞

),

We are allowed to conclude the joint convergence with (Xk)k∈N because of Lemma III.24. Now, because of

the common convergent sequence (Xk)k∈N we can obtain the joint convergence of the two last convergence,i.e.(

Xk, [Y k + Rk,2,I ], 〈Y k + Rk,2,I〉, [Y k − Rk,2,I ], 〈Y k − Rk,2,I〉,Gk) ext−−−−→(

X∞, [Y∞ + R∞,2,I ], 〈Y∞ + R∞,2,I〉, [Y∞ − R∞,2,I ], 〈Y∞ − R∞,2,I〉G∞).

Finally, we conclude the convergence III.44 by Corollary I.122, the polarisation identity and in view ofthe integrability properties provided by Lemma III.23.


III.5. The orthogonal martingale-sequence

Now that we have obtained the families of converging martingales by Proposition III.17 and Proposi-tion III.25, we proceed to collect some properties of the sequence (Nk)k∈N, where Nk has been obtained

by the orthogonal decomposition of Y k with respect to (Gk, Xk,c, µXk,d

); see Theorem III.3. It may seemnaive that we devote a section only for a pair of lemmata, however, we will do so, in order to categorise thepreparatory results in an accessible way. The Subsection III.6.1 will provide some additional informationfor the arbitrary weak limit of (Nk)k∈N.

Lemma III.26. Assume the setting of Theorem III.3. Then, for the sequence (Nk)k∈N the following aretrue:

(i) The sequence (〈Nk〉)k∈Nk is C−tight in D(R).(ii) The sequence (Nk)k∈N is tight in D(R).

(iii) The sequence((Nk, 〈Nk〉)

)k∈N is tight in D(R2).

(iv) The sequence (Nk)k∈N is L2−bounded, i.e. supk∈N E[

supt∈[0,∞] |Nkt |2]<∞.

(v) The sequence (〈Nk〉∞)k∈Nk is uniformly integrable.(vi) The sequence ([Nk]∞)k∈Nk is L1−bounded, i.e. supk∈N E

[[Nk]∞

]<∞.

Proof. (i) The orthogonal decompositions of Y k with respect to (Gk, Xk,c, µXk,d

) and Proposi-tion I.165 yield

〈Zk ·Xk,c + Uk ? µ(Xk,d,Gk), Nk〉 = 0, for every k ∈ N. (III.47)

By (iii)-(iv) of Lemma III.4 we get that the sequence (〈Y k〉)k∈N is C−tight. Moreover, (III.47) impliesthat

〈Y k〉 = 〈Zk ·Xk,c + Uk ? µ(Xk,d,Gk)〉+ 〈Nk〉, for every k ∈ N, (III.48)

which in turn yields that the process 〈Y k〉 strongly majorizes both 〈Zk · Xk,c + Uk ? µ(Xk,d,Gk)〉 and〈Nk〉, for every k ∈ N; see Definition I.144. We can conclude thus the C−tightness of (〈Nk〉)k∈N byProposition I.145.

(ii) Using Theorem I.146 to obtain the tightness of (Nk)k∈N, it suffices to show that (〈Nk〉)k∈N isC−tight and (Nk

0 )k∈N is tight. The first statement follows from (i), while for the second one we get that

Nk0 = 0, by the definition of the orthogonal decomposition of Y k with respect to (Gk, Xk,c, µX

k,d

), forevery k ∈ N.

(iii) This is immediate in view of Lemma I.143 and (i)-(ii).

(iv) We have by Doob’s L2−inequality that

E[

supt∈[0,∞]

∣∣Nkt

∣∣2] ≤ 4E[∣∣Nk

∞∣∣2] = 4E

[〈Nk〉∞

]≤ 4E

[〈Y k〉∞

], (III.49)

where we used Identity (III.48). By Lemma III.4.(v) we obtain that the sequence (〈Y k〉∞)k∈N is uniformlyintegrable and in particular L1−bounded.

(v) Recall (III.48), which implies that (〈Y k〉∞)k∈N strongly majorizes (〈Nk〉∞)k∈N. Moreover, theformer sequence is uniformly integrable, see the comment in the proof of (iv) or Lemma III.4.(v), and byCorollary I.29.(ii) we can conclude.

(vi) The identity E[[Nk]∞] = E

[〈Nk〉∞

]for every k ∈ N and the uniform integrability of the sequence

(〈Nk〉∞)k∈N, which implies its L1−boundedness, allow us to conclude.

Lemma III.27. Let (Nkl)l∈N be a subsequence of (Nk)k∈N and assume that there exists a real-valued

cadlag process N such that Nkl L−−→ N . Then, (Nkl)l∈N is P-UT and, moreover, [Nkl ]L−−→ [N ].

Proof. This is immediate by Proposition I.153 and Lemma III.26.(iv).

III.6. Θ∞, I IS AN F–MARTINGALE, FOR I ∈ J (X∞). 91

III.6. Θ∞, I is an F–martingale, for I ∈ J (X∞).

Let us start this section by returning to the discussion made in Section III.2. We remind the reader(the reader may recall the point after the two bullet points), that our ultimate aim is to identify the

arbitrary weak limit of (Nk)k∈N, named ĎN, with N∞, which is the element of H2,⊥(G∞, X∞,c, μX∞,d

)

given by the orthogonal decomposition of Y ∞ with respect to (G∞, X∞,c, μX∞,d

).Let us deviate for a while from the discussion we started, in order to provide some notation which will

visualise easier our comments. In view of Lemma III.26.(iii) which verifies the tightness of (Nk, 〈Nk〉)k∈N

in D(R2), we will assume throughout the rest of the chapter that the sequence (Nkl , 〈Nkl〉)l∈N is a weaklyconvergent subsequence extracted by an arbitrarily chosen subsequence of (Nk, 〈Nk〉)k∈N. Let us denoteby (ĎN,Ξ) the weak-limit of

((Nkl , 〈Nkl〉))

l∈N, i.e.

(Nkl , 〈Nkl〉) L−−−→l→∞

(ĎN,Ξ), (III.50)

where by Lemma III.26.(i) we know that Ξ is a continuous and increasing process. Having a closerexamine of Convergence (III.50), we will realise that there is some connection between ĎN and Ξ. Indeed,

observe that by the orthogonal decomposition of Y k with respect to (Gk, Xk,c, μXk,d

) we have

〈Y k, Nk〉 = 〈Nk〉 for every k ∈ N. (III.51)

Thus, if the convergence

(Y kl , Nkl)L−−→ (Y ∞, ĎN) in D(R2) (III.52)

is true, then the convergence (III.50) translates to

[Y kl , Nkl ]− 〈Nkl〉 L−−−→l→∞

[Y ∞, ĎN

]− Ξ. (III.53)

Indeed, in view of Lemma III.4.(vi) and Lemma III.27, we need only to prove that (Y kl , Nkl)L−−→

(Y ∞, ĎN) in D(R2) so that convergence (III.53) is valid. Having in (III.53) sufficient integrability for theapproximating martingale sequence, we will finally obtain one more convergent martingale-sequence. Itwill be this convergence with which we will identify Ξ with 〈N∞〉. But let us postpone this discussionuntil Subsection III.6.2.

We return now to the discussion we deviated from. We draw the attention of the reader at the pointwhere we know how to translate the sufficient requirements (Α)-(ΣΤ) into the martingale conditions(III.9)-(III.10), which we state again below for the their convenience:[

R2,I μ(X∞,d,G∞), ĎN

], [X∞,i, ĎN ] ∈ M(F;R), for every i = 1, . . . , (III.9)

and [R2,I μ

(X∞,d,G∞), Y ∞ ]− ⟨R2,I μ(X∞,d,G∞), Y ∞⟩G∞

∈ M(F;R)

[X∞,i, Y ∞]− 〈X∞,i, Y ∞〉G∞ ∈ M(F;R), for every i = 1, . . . , .(III.10)

Pay attention to the fact that we have substituted the integrand h1I with R2,I , where I ∈ J (X∞), i.e.the choice we have done in the previous sections, and that ĎN now denotes the weak limit of (Nk)k∈N

determined by Convergence (III.50). Observe, now, that we have already constructed an approximatingmartingale-sequence for the processes appearing in (III.10), see Section III.4, but this is not the case for(III.9). But, if the convergence

(Xkl ,R2,I μ(Xkl,d,Gkl ), Nkl)

L−−→ (X∞,R2,I μ(X∞,d,G∞), ĎN) in D(R × R× R) (III.54)

holds, we can obtain (III.9), because all the martingale-sequences are P-UT; see Lemma III.5.(v),Lemma III.18, Lemma III.27 and Theorem I.152. Therefore, we know how to approximate every ele-ment in (III.9) - (III.10). The only object that has not been specified is the filtration F. A sufficientlyclear idea on the way we will construct F has been already provided in Section III.2 and we advisethe reader to refresh their memory; see comments for (Α). For this construction we will use Proposi-tion I.147, so now (following the notation of the aforementioned proposition) we have only to clarifyhow the martingale-sequence (Θkl)l∈N

should be chosen. In reality, it is sufficient to collect in Θk all

the k−elements of the convergent Gk−martingale-sequences we will make use of. The following notationdescribes precisely what we tried to explain with words.


Notation III.28. In the following I ∈ J (X∞) and k ∈ N.

• Hk,I :=(Xk, 〈Xk〉Gk

, Y k, 〈Y k〉Gk

, 〈Xk, Y k〉Gk

, Nk, 〈Nk〉,〈R2,I μ

(Xk,d,Gk)〉Gk

, 〈R2,I μ(Xk,d,Gk), Xk〉Gk

, 〈R2,I μ(Xk,d,Gk), Y k〉Gk)

;

• ĎH∞,I :=(X∞, 〈X∞〉G∞

, Y ∞, 〈Y ∞〉G∞, 〈X∞, Y ∞〉G∞

, ĎN,Ξ,

〈R2,I μ(X∞,d,G∞)〉G∞

, 〈R2,I μ(X∞,d,Gk), X∞〉G∞

, 〈R2,I μ(X∞,d,G∞), Y ∞〉Gk)

;

• Hk,I,G := (Hk,I ,E[1G|Gk]), for G ∈ G∞∞ ;

• ĎH∞,I,G := (H∞,I ,E[1G|G∞]), for G ∈ G∞∞ ;

• Θk,I := (Xk, [Xk]− 〈Xk〉Gk

, Nk, [Xk, Nk], [R2,I μ(Xk,d,Gk), Nk], Y k, [Y k, Nk]− 〈Nk〉

[Xk, Y k]− 〈Xk, Y k〉Gk

, [R2,I μ(Xk,d,Gk), Y k]− 〈R2,I μ

(Xk,d,Gk), Y k〉Gk

);

• Θk,I,G := (Θk,I ,E[1G|Gk]), for G ∈ G∞∞ ;

• Θ∞,I := (X∞, [X∞]− 〈X∞〉G∞, ĎN, [X∞, ĎN ], [R2,I μ

(X∞,d,G∞), ĎN ], Y ∞,

[X∞, Y ∞]− 〈X∞, Y ∞〉G∞, [R2,I μ

(X∞,d,G∞), Y ∞]− 〈R2,I μ(X∞,d,G∞), Y ∞〉G∞

);

• Θ∞,I,G := (Θ∞,I ,E[1G|G∞]), for G ∈ G∞∞ ;

• E1 := R × R× × R× R× R × R× R× R× R × R;

• E2 := R × R× × R× R × R× R× R × R

After this discussion, let us summarize what we need to prove so that we can apply Proposition I.147for the subsequence (Θkl,I,G)l∈N:

The family‖Θkl,I,G

t ‖1is uniformly integrable for every I ∈ J (X∞) and for every G ∈ G∞

∞ , whichwill be proven in Lemma III.30.

The Convergence (III.52) and (III.54) are valid for every I ∈ J (X∞). The two aforementionedconvergence can be combined to

(Xkl ,R2,I μ(Xkl,d,Gkl ), Y kl , Nkl)

L−−→ (X∞,R2,I μ(X∞,G∞), Y ∞, ĎN) in D(R × R3), (III.55)

for every I ∈ J (X∞). This is the purpose of Lemma III.31, which provides the tightness of (Hk,I,G,Θk,I,G)k∈N9

in D(E × R), for every I ∈ J (X∞) and for every G ∈ G∞∞ . Observe that on the validity of (III.50), the

tightness of (Θk,I,G)k∈N implies the validity of (III.55).

Definition III.29. For the pair (ĎN,Ξ) as determined by Convergence (III.50), define F := G∞ ∨F(ĎN,Ξ),i.e.

Ft := σ(Gt ∪

ĎNs, s ≤ t

∪ Ξs, s ≤ t)

.

Observe that the filtration F depends on ĎN and Ξ, but we notationally suppressed this dependence.Moreover, it does not necessarily hold that F is right–continuous. However, in view of [27, TheoremVI.3, p. 69] and the cadlag property of Θ∞, we can conclude also that Θ∞ is an F+−martingale, whereF+ := (Ft+)t∈R+ and Ft+ :=

⋂s>t Fs.

Lemma III.30. The family of random variables(∥∥Θk,I,G

t

∥∥1

)k∈N,t∈R+

is uniformly integrable for every

I ∈ J (X∞) and G ∈ G∞∞ .

Proof. Let I ∈ J (X∞) and G ∈ G∞∞ be fixed. Recall that∥∥Θk,I,G

t

∥∥1:=∥∥Xk

t

∥∥1+∥∥[Xk]t − 〈Xk〉t

∥∥1+∣∣Nk

t

∣∣+ ∥∥[Xk, Nk]t‖1+∥∥[Nk,R2,I μ

(Xk,d,G∞)]t∥∥+ ∣∣Y k

t

∣∣+ ∣∣[Y k, Nk]t − 〈Nk〉t∣∣

+∥∥[Xk, Y k]t − 〈Xk, Y k〉t

∥∥1

+∣∣[R2,I μ

(Xk,d,Gk), Y k]t − 〈R2,I μ(Xk,d,Gk), Y k〉Gk

t |+ ∣∣E[1G|Gkt ]∣∣

(III.56)

We are going to prove that for each summand of ‖Mk,I,Gt ‖1 the associated family indexed by k ∈ N, t ∈

[0,∞] is uniformly integrable. Then, we can conclude also for their sum (‖Mk,I,Gt ‖1)k∈N,t∈[0,∞] by

Corollary I.29.(iv).Let us first recall that the following families are uniformly integrable

9With the exact same arguments we can prove that (Hk,I,G)k∈Nis tight in D(E×R). For our convenience, we preferred

to state Lemma III.31 for the martingale sequence.


(Tr[[Xk]∞

])k∈N,

(Tr[〈Xk〉∞

])k∈N

([Y k]∞

)k∈N,

(〈Y k〉∞

)k∈N,

([

R2,I ? µ(Xk,d,Gk)

]∞

)k∈N,

(⟨R2,I ? µ

(Xk,d,Gk)⟩∞

)k∈N and

(〈Nk〉∞)k∈N,

see Lemma III.5.(iv), Lemma III.4.(v), Corollary III.16 and Lemma III.26.(v), while by Lemma III.26.(iv)we obtain that

([Nk]∞)k∈N is L1−bounded.

Now, in view of the following relationships

supk∈N,t∈R+

E[|Y kt |2

]≤ supk∈N,t∈R+

E[[Y k]∞

]<∞, sup

k∈N,t∈R+

E[|Nk

t |2]≤ supk∈N,t∈R+

E[[Nk]∞

]<∞

and

supk∈N,t∈R+

E[|Xk,i

t |2]≤ supk∈N,t∈R+

E[[Xk]ii∞

]≤ supk∈N,t∈R+

E[Tr[[Xk]∞

]]<∞ for every i = 1, . . . , `,

we conclude by the de La Vallee Poussin Theorem, see Theorem I.27, the uniform integrability of thefamilies

‖Xk

t ‖1, l ∈ N and t ∈ R+,|Y kt |, l ∈ N and t ∈ R+ and

|Y kt |, l ∈ N and t ∈ R+.

The family E[1G|Gk], k ∈ N and t ∈ R+ is uniformly integrable by Corollary I.29.(iii). Finally, forthe rest of the summands we apply either Lemma A.2 or Lemma A.3 of Appendix A.2, in order to obtainthe uniform integrability of the families

(‖Var

([Xk, Nk]

)∞‖1

)k∈N,

(‖Var

([Xk, Y k]

)∞‖1

)k∈N,

(‖Var

(〈Xk, Y k〉

)∞‖1

)k∈N,

(Var([R2,I ? µ

(Xk,d,Gk), Nk])∞

)k∈N,

(Var(〈R2,I ? µ

(Xk,d,Gk), Nk〉)∞

)k∈N and

(Var([Y k, Nk]

)∞

)k∈N,

This proves our claim that each summand is uniformly integrable.

Lemma III.31. The sequence(Hk,I,G,Θk,I,G

)k∈N is tight in D(E1×R×E2×R) for every I ∈ J (X∞)

and for every G ∈ G∞∞ .

Proof. Let I ∈ J (X∞) and G ∈ G∞∞ be fixed. We claim that the sequence Φk := (Xk, Y k, Nk) istight in D(R` × R2), and that the tightness of (Φk)k∈N is sufficient to show the tightness of (Θk,I,G)k∈N.For the following we are going to prove for simplicity that (Θk,I,G)k∈N is tight in D(E2×R), since (simpleror) analogous arguments prove the tightness of (Hk,I,G)k∈N in D(E1 × R).

Let us argue initially for our second claim. The space D(E×R) is Polish. Hence in this case tightnessis equivalent to sequential compactness. Therefore, it suffices to provide a weakly convergent subsequencefor every subsequence of (Θk,I,G)l∈N. To this end, let us consider a subsequence (Θkm,I,G)m∈N. Assumingthe tightness of (Φk)k∈N, there exists a weakly convergent in D(R`+2) subsequence (Φkmn )n∈N, converging

to, say, (X∞, Y∞, N). Moreover, we remind the reader that the following sequences are P-UT

(Xk)k∈N; by Lemma III.5.(v). Hence also the subsequence (Xkmn )n∈N. (Y k)k∈N; by Lemma III.4.(vi). Hence also the subsequence (Y kmn )n∈N. (Nkmn )n∈N; in view of the L2−boundedness of the martingale-sequence (Nk)k∈N and the conver-

gence NkmnL−−→ N apply Lemma III.27.

(R2,I ? µ(Xk,d,Gk))k∈N; see Lemma III.18. Hence also the subsequence (R2,I ? µ

(Xkmn,d,Gkmn ))m∈N.

Then, by Theorem I.152, Proposition III.17 and the above

(Xkmn , Y kmn , Nkmn , [Xkmn , Nkmn ], [Y kmn , Nkmn ], [R2,I ? µ(Xkmn,d,Gkmn ), Nkmn ])yL

(X∞, Y∞, N∞, [X∞, N∞], [Y∞, N∞], [R2,I ? µ(X∞,d,G∞), N∞])

Moreover, recall the following convergence:

(Xk, [Xk], 〈Xk〉) (J1(R`×R`×`×R`×`),P)−−−−−−−−−−−−−−−−−→ (X∞, [X∞], 〈X∞〉G∞

),

by Lemma III.5, and(Xk, Y k, [Y k, Xk], 〈Y k, Xk〉

) (J1(R`×R×R`×R`),P)−−−−−−−−−−−−−−−→

(X∞, Y∞, [Y∞, X∞], 〈Y∞, X∞〉

); (III.6)


(Xk,

[Y k,R2,I μ

(Xk,d,Gk)]− ⟨Y k,R2,I μ

(Xk,d,Gk)⟩) (J1(R

×R),L2×L1)−−−−−−−−−−−−−→(

X∞,[Y k,R2,I μ

(Xk,d,Gk)]− ⟨Y k,R2,I μ

(Xk,d,Gk)⟩).(III.44)

Hence, by Theorem I.152 and Corollary I.125, (this is the reason we actually needed for all the convergencethe joint convergence with (Xk)k∈N) we also get the convergence in D(E)(Xkmn , [Xkmn ]− 〈Xkmn 〉, Nkmn , [Xkmn , Nkmn ], [R2,I μ

(Xklm

,d,Gkmn ), Nkmn ], Y kmn ,

[Y kmn , Nkmn ], [Xkmn ,Y kmn], [R2,I μ

(Xklm

,d,Gkmn ), Y kmn ]− 〈R2,I μ

(Xklm

,d,Gkmn ), Y kmn 〉);L(

X∞, [X∞]− 〈X∞〉, N , [X∞, N ], [R2,I μ(X∞,d,G∞), N ], Y ∞,

[Y ∞, N ], [X∞, Y ∞], [R2,I μ(X∞,d,G∞), Y ∞]− 〈R2,I μ

(X∞,d,G∞), Y ∞〉).In view of the C−tightness of (〈Nk〉)k∈N, see Lemma III.26, we can pass to a further subsequence -whichwe will denote again by (kmn

)n∈N- so that

〈Nkmn 〉 L−−−−→n→∞ Ξ.

Hence, we can finally obtain a jointly weakly convergent in D(E × R)

Θkmn ,I,G;L(X∞, [X∞]− 〈X∞〉, N , [X∞, N ], [R2,I μ

(X∞,d,G∞), N ], Y ∞, [Y ∞, N ]− Ξ,

[X∞, Y ∞], [R2,I μ(X∞,d,G∞), Y ∞]− 〈R2,I μ

(X∞,d,G∞), Y ∞〉,E[1G|G∞· ]),

where we have used that (Xk,E[1G|Gk· ])

(J1(R+1),P)−−−−−−−−−−→

k→∞(X∞,E[1G|G∞

· ]) to conclude.

In order to prove our initial claim that (Φk)k∈N is tight, we will apply Theorem I.146 to theL2−bounded sequences (Xk)k∈N, (Y

k)k∈N, and (Nk)k∈N. The sequences (Xk0 )k∈N, (Y

k0 )k∈N, and (Nk

0 )k∈N

are clearly tight in R, R and R respectively, as L2−bounded. The sequences(Tr[〈Xk〉])

k∈N, (〈Y k〉)k∈N

and (〈Nk〉)k∈N are C−tight; see respectively Lemma III.5, Lemma III.4 and Lemma III.26.(i). Finally,the sequence (Ψk)k∈N, where

Ψk := Tr[〈Xk〉]+ 〈Y k〉+ 〈Nk〉,

is C−tight as the sum of C−tight sequences; see Lemma I.143. This concludes the proof.

Remark III.32. In the proof of the previous lemma, in order to prove the tightness of the sequence Φk

we have used Theorem I.146, which in turn makes use of Aldous’ criterion for tightness; see [41, SectionVI.4a]. This allows us to conclude that every weak–limit point of (Nk)k∈N is quasi–left–continuous forits natural filtration. Recall that by Condition (M1) we have in particular that X∞ and Y ∞ are quasi–left–continuous for their natural filtrations. To sum up, for the arbitrary weak–limit point ĎN of (Nk)k∈N,it holds

P(ΔX∞,it = 0) = P(ΔY ∞

t = 0) = P(ΔĎNt = 0) = 1, for every t ∈ R+, i = 1, . . . , .

Remark III.33. In the rest of the section we will use the pair (ĎN,Ξ) obtained by (III.50). Thiscan be done without loss of generality, since the pair (ĎN,Ξ) was a weak-limit of an arbitrarily chosensubsequence of

((Nk, 〈Nk〉)

k∈N. At this point we introduce the following convention for the rest of the

chapter: whenever the subsequence (kl)∈N is used, it will be understood as the sequence of N which isdetermined by the convergence (III.50).

Corollary III.34. For every I ∈ J (X∞), G ∈ G∞∞ , we have that Θkl,I,G L−−−→

l→∞Θ∞,I,G and that the

process Θ∞,I,G is a uniformly integrable FĎH∞,I,G−martingale.

Proof. Let us fix an I ∈ J (X∞) and a G ∈ G∞∞ . By Lemma III.31, i.e. by the tightness in D(E×R)

of the sequence (Θk,I,G)k∈N, we obtain that the sequence (Θkl,I,G)l∈N is tight in D(E×R). Therefore, it issufficient to prove the convergence in law of each element of the subsequence (Θkl,I,G)l∈N. We will abstainfrom providing all of them in detail, since we will use exactly the same arguments as in Lemma III.31,but for the weakly convergent subsequence (Θkl)l∈N with weak limit Θ∞,I,G.


In order to show the second statement, we have only to recall Lemma III.30 and that Θkl,I,G isE × R−valued Gkl−martingale for every l ∈ N. Now we can conclude that Θ∞,I,G is a uniformlyintegrable martingale with respect to its natural filtration by Proposition I.147. Finally, in order to

obtain the martingale property with respect to FĎH∞,I,G

, i.e. the usual augmentation of the naturalfiltration of Θ∞,I,G, we apply [27, Theorem VI.3, p. 69].

At this point we remind the reader some properties which we will make use in the coming proposition.

Remark III.35. (i) For every t ∈ [0,∞] it holds Ft = G∞t ∨ F (ĎN,Ξ)

t .

(ii) The inclusion FĎH∞,I,G ⊂ F holds for every fixed G ∈ G∞

∞ . In particular, for every fixed G ∈ G∞∞ ,

every FĎH∞,I,G−predictable process is also F−predictable.

Proposition III.36. The process Θ∞,I is a uniformly integrable F−martingale, for every I ∈ J (X∞).

Proof. Let us fix an I ∈ J (X∞). By applying Corollary III.34 for every G ∈ G∞∞ we have that

Θ∞,I is a uniformly integrable FĎH∞,I,G−martingale. Therefore we have only to prove that actually Θ∞,I

is also an F−martingale. By Lemma A.7, it is sufficient to prove that for every 0 ≤ t < u ≤ ∞ thefollowing condition holds ∫

Λ

E[Θ∞,I

u

∣∣Ft

]dP =

∫Λ

Θ∞,It dP, (III.57)

for every Λ ∈ At :=Γ∩Δ |Γ ∈ G∞

t ,Δ ∈ F (ĎN,Ξ)t

. The class At is indeed a π−system10 with σ(At) = Ft.

Let us, therefore, fix 0 ≤ t < u ≤ ∞ and Λ ∈ At, where Λ = Γ ∩Δ, for some Γ ∈ G∞t , Δ ∈ F (ĎN,Ξ)

t .

Observe that in particular Λ ∈ FĎH∞,I,Γ

t . Now we obtain∫Λ

E[Θ∞,I

u

∣∣Ft

]dP =

∫Λ

E

[E[Θ∞,I

u

∣∣Ft

]∣∣∣FĎH∞,I,Γ

t

]dP =

∫Λ

E

[Θ∞,I

u

∣∣∣FĎH∞,I,Γ

t

]dP =

∫Λ

Θ∞,It dP, (III.58)

where the first equality holds because Λ ∈ FĎH∞,I,Γ

t , therefore we use the definition of the conditional

expectation with respect to the σ−algebra FĎH∞,I,Γ

t . In the second equality we have used the tower

property and Remark III.35.(ii), while for the third one we have used that Θ∞,I is a FĎH∞,I,Γ−martingale;

we used Corollary III.34 for Γ ∈ G∞t ⊂ G∞

∞ . Therefore, we can conclude that Θ∞,I is a uniformly integrableF−martingale.

III.6.1. ĎN is sufficiently integrable. The uniform integrability of the F−martingale ĎN implies

neither the integrability of [ ĎN ]12∞ nor of supt∈[0,∞) |ĎNt|. In Lemma III.39 we will prove that there exists

a suitable Ψ ∈ YFmod whose Young conjugate Ψ is also moderate and such that Ψ(supt∈[0,∞)] |ĎNt|) andΨ([ ĎN ]

12∞)are integrable.11 This result is crucial in showing that MμX∞

[ΔĎN

∣∣PF]is well–defined, i.e.

|ΔĎN |μX∞,d ∈ Aσ(F); see Corollary III.40 and Proposition III.41.Since one may wonder about the necessity of such a function Ψ, let us clarify the situation. Let us

first recall the notation of Example I.46, where we have defined Υϑ(x) :=1ϑx

ϑ, for every ϑ ∈ [1,∞]. Now,

in view of the L2−boundedness of (Nk)k∈N, recall Lemma III.26, we can prove Lemma III.39 for Υϑ, forevery ϑ ∈ [1, 2). If, moreover, ϑ ∈ (1, 2), then the Young conjugate of Υϑ is Υϑ , with ϑ ∈ (2,∞) whichis also moderate. However, if we were to prove Corollary III.40, we should have assumed the uniform

integrability of the sequence(Tr[[X∞]

ϑ

2∞])

k∈N. But this condition is equivalent to the convergence

Xk∞

‖·‖Υϑ−−−−−−→ X∞∞ for some ϑ ∈ (2,∞), 12

which is stronger than the one assumed to (M2). Recall that in (M2) we have assumed the convergenceof the terminal random variables to hold under ‖ · ‖L2(G;R).

The choice of the function Ψ which we are going to utilise is given in Definition III.37. However, beforewe reach at that point, let us comment on the way we chose this function. We have shown in Lemma III.14that the sequence

(Tr[[Xk]∞

])k∈N

is uniformly integrable. The orthogonal decomposition of Y ∞ with

10A π−system is a non–empty family of sets which is closed under finite intersections.11It is this specific point that we had to obtain Proposition I.51 in Subsection I.2.2.12See Notation I.52. Actually, this is the convergence under the norm of the classical Lϑ−space.


respect to (G∞, X∞,c, μX∞,d

) implies then that the (finite) family 〈Z∞ ·X∞,c〉∞, [U∞ μ(X∞,d,G∞)]∞is uniformly integrable. Therefore, the family

U :=Tr[[Xk]∞

], k ∈ N

∪ 〈Z∞ ·X∞,c〉∞, [U∞ μ(X∞,d,G∞)]∞

(III.59)

is uniformly integrable as a finite union of uniformly integrable families of random variables; see Corol-lary I.29.(iv). By the de La Vallee Poussin–Meyer Criterion, see Corollary I.48, there exists a moderateYoung function with associated moderate sequence A(U) such that

supΓ∈U

E[ΦA(U)(|Γ|)] < ∞. (III.60)

Since the moderate sequence A(U) for which (III.60) is satisfied is not unique, we will hereinafter fix oneand we will denote it A(U) as well.Notation III.37. For the rest of the chapter ΨU := (ΦA(U)quad), where ΦA(U) is the moderate Youngfunction for which (III.60) holds. In other words, ΨU is the Young conjugate of ΦA quad.Lemma III.38. It holds ΦA(U),ΨU ∈ YFmod, which is equivalent to

(scΨU )

= cΦA(U)quad

, (cΨU

) = scΦA(U)quad

and all the constants are finite. Moreover, there exists ΥU ∈ YF such that ΥU ΨU = quad.

Proof. See Proposition I.51 and Corollary I.45.

Lemma III.39. The weak–limit ĎN given by convergence (III.50) and the Young function ΨU fromNotation III.37 satisfy

E

[ΨU(

supt∈[0,∞)

|ĎNt|)]

< ∞ and E

[ΨU([ ĎN ]

12∞)]

< ∞.

Proof. By Convergence (III.50), Corollary III.34, Remark III.32 and Proposition I.140 we have that

Nklt

L−−−→l→∞

ĎNt, for every t ∈ R+.

The function ΨU is continuous and convex, since it admits a representation via a Lebesgue integral withpositive and non-decreasing integrand. By the continuity of ΨU and the above convergence we can alsoconclude that

ΨU (|Nklt |) L−−−→

l→∞ΨU (|ĎNt|), for every t ∈ R+. (III.61)

By Lemma III.38, there exists a Young function ΥU such that

supl∈N

E

[ΥU(ΨU(

supt∈[0,∞)

|Nklt |))] = sup

l∈N

E

[sup

t∈[0,∞)

|Nklt |2] Lem. III.26.(iv)

< ∞. (III.62)

By the above equality and de La Vallee Poussin Theorem, see Theorem I.27, we obtain the uniformintegrability of the family

(ΨU (|Nkl

t |))l∈N,t∈[0,∞)

. On the other hand, convergence (III.61) and the

Dunford–Pettis Compactness Criterion, see Theorem I.26, yield that the set

Q :=ΨU (|Nkl

t |), t ∈ [0,∞), l ∈ N ∪ ΨU (|ĎNt|), t ∈ [0,∞)

is uniformly integrable, since we augment the relatively weakly compact set(ΨU (|Nkl

t |))t∈[0,∞),l∈N

merely

by aggregating it with the weak–limits ΨU (|ĎNt|), for t ∈ [0,∞). In particular, the subset ĎNΨU :=(ΨU (|ĎNt|)

)t∈[0,∞)

is uniformly integrable and in particular L1−bounded; see Theorem I.25. Now, the

L1−boundedness of ĎNΨU and the fact that ĎN is a uniformly integrable F−martingale, see PropositionIII.36, imply that the random variable ĎN∞ := limt↑∞ ĎNt exists P−almost surely, see Theorem I.57. Usingthe uniform integrability of ĎNΨU and the continuity of ΨU once again, we have that

ΨU (|ĎNt|) L1(G;R)−−−−−−−→t↑∞

ΨU (|ĎN∞|), (III.63)

i.e. ΨU (|ĎN∞|) ∈ L1(G;R); see Vitali’s Convergence Theorem which is stated as Theorem I.32. Recallnow that the function ΨU is moderate and convex, see Lemma III.38 and recall Definition I.37. By the

integrability of ΨU (|ĎN∞|) we have that ‖ĎN∞‖ΨU < ∞, where ‖Θ‖ΨU := infλ > 0,E

[ΨU( |Θ|

λ

)] ≤ 1is

the norm of the Orlicz space associated to the Young function ΨU , see Proposition I.53.


Now we are ready to apply Doob’s Inequality in the form of Theorem I.104 since the Young conjugateof ΨU is ΦA(U) quad ∈ YFmod with associated constant (scΨU )

< ∞; see Lemma III.38. The aboveyield ∥∥∥ sup

t∈R+

|ĎNt|∥∥∥ΨU

≤ (cΨU

)∥∥ĎN∞

∥∥ΨU

< ∞. (III.64)

Inequality (III.64) yields therefore the finiteness of ‖ supt∈[0,∞) |ĎNt|‖ΨU , which in conjunction with the

fact that ΨU is moderate, i.e. cΨU < ∞, delivers also the finiteness of E[ΨU (supt∈[0,∞) |ĎNt|)]; see

Proposition I.53.(v).Finally, we make use of the finiteness of E

[ΨU (supt∈[0,∞) |ĎNt|)

], of the Burkholder–Davis–Gundy

Inequality, Theorem I.101, and of the fact that ΨU ∈ YFmod to conclude that

E[ΨU([ ĎN ]

12∞)]

< ∞.

At this point, in order not to disrupt the flow of results regarding ĎN , we have to anticipate fora while in the next subsection. In particular we will convey the main message of Corollary III.44and Lemma III.45, which state that the F−martingale X∞ is F−quasi-left-continuous and such that

ν(X∞,d,F)|PG∞ = ν(X

∞,d,G∞). Now, we can proceed to the next

Corollary III.40. The weak–limit ĎN in convergence (III.50) satisfies

E

[ ∫(0,∞)×R

|ΔĎNs| ‖x‖1 μX∞,d

(ds, dx)]< ∞.

In other words, |ΔĎN |μX∞,d ∈ Aσ(F); recall for this Definition I.75. Moreover, the processes [Y ∞, ĎN ],

〈Z∞ ·X∞,c, ĎN c〉, [U∞ μ(X∞,d,G∞), ĎNd] and [X∞, ĎN ] are of class (D) for the filtration F.

Proof. We prove initially that the required expectation is indeed finite.

E

[ ∫(0,∞)×R

|ΔĎNs| ‖x‖1 μX∞,d

(ds, dx)]=

∑i=1

E

[ ∫(0,∞)×R

|ΔĎNs| |xi|μX∞,d

(ds, dx)]

=∑

i=1

E

[∑s>0

|ΔĎNs| |ΔX∞,d,is |

]≤

∑i=1

E

[(∑s>0

(ΔĎNs)2) 1

2(∑s>0

(ΔX∞,d,is )2

) 12

]

≤∑

i=1

E

[[ ĎN ]

12∞ [X∞,i]

12∞] Lem. I.44.(i)

≤Not. III.37

∑i=1

E

[ΨU([ ĎN ]

12∞)+ΦA(U) quad

([X∞,i]

12∞)]

≤ E

[ΨU([ ĎN ]

12∞)+ΦA(U)

(Tr[[X∞]∞

])] (III.60)<

Lemma III.39∞,

where in the last inequality we used also the convexity of ΦA(U) in order to take out the coefficient 12 ,

which appears due to the definition of the function quad.We are going to prove the second statement only for [Y ∞, ĎN ]. It is sufficient to prove for the associated

total variation process that Var([Y ∞, ĎN

])∞ ∈ L1(Ω,G,P). For the following calculation we are going to

use Property (III.60) and Lemma III.39, exactly as we did above.

Var([Y ∞, ĎN

])∞ = Var

([Z∞ ·X∞,c + U∞ μ(X∞,d,G∞), ĎN

])∞

≤ Var([Z∞ ·X∞,c, ĎN

])∞ +Var

([U∞ μ(X∞,d,G∞), ĎN

])∞

= Var(〈Z∞ ·X∞,c, ĎN c〉)∞ +Var

([U∞ μ(X∞,d,G∞), ĎNd]

)∞

KW Ineq.

≤ 〈Z∞ ·X∞,c〉 12∞〈ĎN c〉 1

2∞ + [U∞ μ(X∞,d,G∞)]12∞[ĎNd] 1

2

∞

≤ 〈Z∞ ·X∞,c〉 12∞[ĎN] 1

2

∞ + [U∞ μ(X∞,d,G∞)]12∞[ĎN] 1

2

∞

Lem. I.44.(i)

≤ [ΦA(〈Z∞ ·X∞,c〉∞

)+ΦA

([U∞ μ(X∞,d,G∞)

]∞)]ΨU([

ĎN] 1

2

∞) (III.60)

<Lem. III.39

∞.


We conclude this sub–sub–section with the following result.

Proposition III.41. The weak–limit ĎN of convergence (III.50) satisfies

〈X∞,c,i, ĎN c〉F = 0 for every i = 1, . . . and MμX∞,d [ΔĎN |PF] = 0. (III.65)

Proof. We will apply Corollary III.8 to the pair of F−martingales (X∞, ĎN). The following verifythat the the requirements of the aforementioned corollary are indeed satisfied.

(i) By Definition III.29 holds G∞ ⊂ F.(ii′) By (M2) holds X∞ ∈ H2(G∞;R) and by Lemma III.39 we have that ĎN ∈ M(F;R). By Propo-

sition III.36 we have that X∞ is also an F−martingale.(iii) The function R2,I satisfies the required properties.(iv) This is Lemma III.45.(v′) This is Corollary III.40. Moreover, by Proposition III.36 we have that [X∞,i, ĎN ] is an F−martingale

for every i = 1, . . . , .(vi′) By Lemma A.6 we have that σ

(J (X∞))= B(R). Moreover, Proposition III.36 verifies that[

R2,I μ(Xd,G∞), Y ∞] ∈ M(F;R) for every I ∈ J (X∞).

(vii) This is Corollary III.40.

The above verify our claim. III.6.2. The orthogonal decomposition of Y ∞ is not affected. In this sub–sub–section we use

the notation and framework of Section III.6. Recall that the filtration F has been defined in (III.29), andthat the subsequence (kl)l∈N is fixed and such that convergence (III.50) holds.

Lemma III.42. Let X0+X∞,c,F+Idμ(X∞,d,F) be the canonical representation of X∞ as F−martingale

and X0 +X∞,c + Id μ(X∞,d,G∞) be the canonical representation of X∞ as G∞−martingale. Then the

respective parts are indistinguishable, i.e.

X∞,c = X∞,c,F and Id μ(X∞,d,G∞) = Id μ

(X∞,d,F) up to indistinguishability.

Therefore, we will simply denote the continuous part of the F−martingale X∞ by X∞,c and we will

use indifferently Id μ(G∞,X∞,c,μX∞,d

) and Id μ(F,X∞,c,μX∞,d

) to denote the discontinuous part of theF−martingale X∞.

Proof. By Proposition III.36 the process X∞ is an F−martingale whose canonical representationis given by

X∞ = X0 +X∞,c,F + Id μ(X∞,d,F),

see [41, Corollary II.2.38]. However X∞ is G∞−adapted, which in conjunction with [34, Theorem 2.2]

implies that the process X∞,c,F + Id μ(X∞,d,F) is a G∞−martingale. On the other hand, the canonical

representation of the process X∞ as a G∞−martingale is

X∞ = X0 +X∞,c + Id μ(X∞,d,G∞).

Hence, by Theorem I.6 we can conclude the indistinguishability of the respective parts due to the unique-ness of the decomposition. Lemma III.43. The following holds 〈X∞〉G∞

= 〈X∞〉F.Therefore, we will denote the dual predictable projection of X∞ simply by 〈X∞〉.Proof. In Proposition III.36 we showed that the process [X∞]− 〈X∞〉G∞

is an F−martingale. Onthe other hand,

(Tr[[X∞]t

])t∈[0,∞]

is uniformly integrable and in particular of class (D) for the filtration

F since the process is increasing. Consequently, by Theorem I.64, there exists a unique F−predictableprocess, say 〈X∞〉F, such that [X∞] − 〈X∞〉F is a uniformly integrable F−martingale. Recall that bydefinition G∞ ⊂ F, therefore PG

∞ ⊂ PF, which allows us to conclude that 〈X∞〉G∞ − 〈X∞〉F is auniformly integrable F−predictable F−martingale of finite variation. Therefore, by Corollary I.65 weobtain that 〈X∞〉G∞ − 〈X∞〉F = 0 up to an evanescent set. Corollary III.44. The process X∞ is F−quasi–left–continuous. Therefore,

ν(X∞,d,F)(ω; t × R) = 0 for every (ω, t) ∈ Ω× R+. (III.66)


Proof. The F−quasi–left–continuity of X∞ is immediate by the continuity of 〈X∞〉 and Theo-rem I.62. By Proposition I.87 we conclude also (III.66).

Now, we can obtain some useful properties about the predictable quadratic covariation of the con-tinuous and the purely discontinuous martingale part of X∞.

Lemma III.45. The following hold 〈X∞,c〉F = 〈X∞,c〉G∞ and ν(X∞,d,F)|PG∞ = ν(X∞,d,G∞).

Therefore, we will denote the dual predictable projection of X∞,c simply by 〈X∞,c〉.

Proof. For the reader’s convenience we separate the proof in two parts.

(i) Firstly we prove that ν(X∞,d,F)|PG∞ = ν(X∞,d,G∞). Indeed, X∞ is both G∞− and F−quasi–left–continuous, it holds P−almost surely for every t ∈ R+∫

R`x µ(X∞,d,G∞)(t × dx)

(M1)=

∫R`xµX

∞,d(t × dx)

(III.66)=

∫R`x µ(X∞,d,F)(t × dx). (III.67)

Consequently, for every non–negative, G∞–predictable function θ, using Theorem I.79, it holds

E[θ ∗ ν(X∞,d,F)

∞] (III.67)

= E[∑s>0

θ(s,∆X∞,d)1[∆X∞,d 6=0]

]= E

[θ ∗ ν(X∞,d,G∞)

∞].

Therefore we can conclude that ν(X∞,d,F)∣∣PG∞ = ν(X∞,d,G∞).

(ii) Now we prove that 〈X∞,c〉G∞ = 〈X∞,c〉F. We will combine the previous part with Lemma III.43.The map Id` is both G∞− and F−predictable function as deterministic and continuous. Then

〈X∞,c〉F = 〈X∞〉 − 〈Id` ? µ(X∞,d,F)〉F Lem. III.42=

Cor. III.44〈X∞〉 − |Id`|2 ∗ ν(X∞,d,F)

(i)= 〈X∞〉 − |Id`|2 ∗ ν(X∞,d,G∞) (M1)

= 〈X∞〉 − 〈Id` ? µ(X∞,d,G∞)〉G∞

= 〈X∞,c〉G∞.

In view of the previous lemmata, we are able to prove that every G∞−stochastic integral with respectto X∞ is also an F−martingale. The exact statement is provided in the next results.

Lemma III.46. For every Z ∈ H2(G∞, X∞,c;R`) holds Z ·X∞,c ∈ H2,c(F;R)

Proof. By Lemma III.45 we can easily conclude that H2(G∞, X∞,c;R`) ⊂ H2(F, X∞,c;R`). We aregoing to prove the required property initially for simple integrands. Assume that ρ, σ are G∞−stoppingtimes such that ρ ≤ σ P − a.s. and that ψ is an R`−valued, bounded and G∞ρ −measurable randomvariable. Then, see Theorem I.67.(i), the G∞−stochastic integral (ψ1Kρ,σK) ·X∞,c is defined as

ψ1Kρ,σK ·X∞,c =∑i=1

∫ ·0

ψi1Kρ,σKdX∞,c,i. (III.68)

Treating now ψ as an Fρ−measurable variable, since G∞ ⊂ F, and using that X∞,c is an F−martingale,Proposition III.36, yields that the representation in (III.68) is also an F−martingale.

Let now Z ∈ H2(G∞, X∞,c;R`) such that Z is bounded. In this case Z ·X∞,c =∑ì=1 Z

i·X∞,c,i. But,using classical arguments, we can approximate Zi, i = 1, . . . , `, by a sequence of simple G∞−predictableprocesses, name (Zk,i)k∈N, i = 1, . . . , `. On the other hand, using the previous step we obtain thatZk,i · X∞,c,i ∈ H2,c(F;R), as linear combination of elements of H2,c(F;R). Therefore, Zi · X∞,c,i ∈H2,c(F;R), for i = 1, . . . , `, as limit of square-integrable F−martingales. Thus, also Z ·X∞,c ∈ H2,c(F;R).

Finally, by Theorem I.67 we can conclude for an arbitrary Z ∈ H2(G∞, X∞,c;R), since the processZ ·X∞,c can be approximated by

((Z1[‖Z‖2≤k]) ·X∞,c

)k∈N ⊂ H

2,c(F;R).

Lemma III.47. For every U ∈ H2(G∞, µX∞,d ;R) holds U ? µ(X∞,d,G∞) ∈ H2,d(F;R). Moreover, the

processes U ? µ(X∞,d,G∞), U ? µ(X∞,d,F) are indistinguishable.

Proof. Let U ∈ H2(G∞, µX∞,d ;R). The inclusion PG∞ ⊂ PF and the equalities

E[∑s>0

∣∣∣ ∫RÙ(s, x)µ(F,X∞,c,µX

∞,d)(s × dx)

∣∣∣2] Lemma III.42= E

[∑s>0

∣∣∣ ∫RÙ(s, x)µX

∞(s × dx)

∣∣∣2](M1)

= E[∑s>0

∣∣∣ ∫RÙ(s, x)µ(G∞,X∞,c,µX

∞,d)(s × dx)

∣∣∣2] <∞yield that U ∈ H2(µX

∞,d,F;R), hence the F−martingale U ? µ(X∞,d,F) is well-defined.


The crucial property for the following comes from Lemma III.45. More precisely, we exploit the factthat

ν(X∞,d,F)|PG∞ = ν(X∞,d,G∞)

in order to obtain from Theorem I.90 that for A ∈ PG∞ the (local) G∞−martingales 1A ? µ(X∞,d,G∞),

1A ? µX(∞,d,F) are indistinguishable. Therefore, in view of the (local) F−martingale property of the latter

process, we can conclude also that 1A ? µ(X∞,d,G∞) is a (local) F−martingale.

Now, the arguments we are going to use are similar to Lemma III.46. Let us initially fix an i ∈1, . . . , ` and assume that a U i is a function of the form

U i(ω, s, x) := ξi(ω)1(ρi1(ω),ρi2(ω)](s)1Ii(x)πi(x), (III.69)

where Ii ∈ B(R`), ρi1, ρi2 are G∞−predictable times and ξi is a G∞ρi1−measurable and bounded random

variable. Recall that πi denotes the i−canonical projection R` 3 x 7→ xi ∈ R. It is immediate that U i isa G∞−predictable function. Observe that for every G∞−stopping time τ

∆(U i ? µX

(∞,d,G∞))τ

= U(τ,∆X∞,dτ )(III.69)

= ξi1Ii(∆X∞,dτ )∆X∞,d,iτ ,

i.e. we can easily conclude that U i ∈ H2(G∞, µX∞,d ;R). Then, we have that U i ? µ(X∞,d,G∞) and

U i ? µ(X∞,d,F) are well-defined and the former lies in H2,d(G∞;R) while the latter lies in H2,d(F;R). Wewill prove now that they are indistinguishable. In view of the above discussion and Proposition I.86, wehave that

U i ? µX(∞,d,G∞) = (ξi1(ρi1,ρ

2i ]πi) ·

(1Ii ? µ

(X∞,d,G∞))

= (ξi1(ρi1,ρ2i ]πi) ·

(1Ii ? µ

(X∞,d,F))

= U i ? µX(∞,d,F). (III.70)

The Identity (III.70) allows us to conclude that U i ? µX(∞,d,G∞) ∈ H2,d(F;R).

Now we let i run over 1, . . . , ` and define

U(ω, s, x) :=∑i=1

ξi(ω)1(ρi(ω),ρi+1(ω)](s)1Ii(x)πi(x), (III.71)

where Ii ∈ B(R`) for every i = 1, . . . , `, (ρi)i=1,...,` is a finite family of G∞−predictable times and ξi isa G∞ρi−measurable and bounded random variable for every i ∈ 1, . . . , `, we can conclude by linearity

that U ? µ(X∞,d,G∞) ∈ H2,d(F;R).

The case U ∈ H2(G∞, µX∞,d ;R) and U is bounded can be concluded by utilising the usual approxi-

mation arguments. For the general case we have the following. Let U ∈ H2(G∞, µX∞,d ;R) and observe

that the sequence (Uk)k∈N, where Uk := U1[ 1k≤|U |≤k] for k ∈ N, is such that Uk?µ(X∞,d,G∞) ∈ H2,d(F;R).

Moreover,

W k := [U ? µ(X∞,d,G∞) − Uk ? µ(X∞,d,G∞)] =∑s>0

(U(s,∆X∞,ds )1[ 1

k≤|U |≤k]

)2

−−−−→k→∞

0 P− a.s..

Using the fact that 0 ≤ W k ≤ [U ? µ(X∞,d,G∞)]∞, where [U ? µ(X∞,d,G∞)]∞ ∈ L1(G;R), we have thatthe sequence (W k) − k ∈ N is uniformly integrable. Thus, we apply Vitali’s Theorem to conclude that

the above convergence holds in L1, which in turn implies the L2−convergence of(Uk ? µ

(X∞,d,G∞)∞

)k∈N

to U ? µ(X∞,d,G∞)∞ . This allows to conclude that the limit U ? µ(X∞,d,G∞) ∈ H2,d(F;R) as limit of square-

integrable F−martingales.

Proposition III.48. For the F−martingale Y∞ holds

〈X∞,c,i, Y∞,c〉F = 〈X∞,c,i, Y∞,c〉G∞

for every i = 1, . . . `

and

MµX∞,d [∆Y∞|PF] = M

µX∞,d [∆Y∞|PG∞ ].

Proof. We will apply Proposition III.7 to the pair (X∞, Y∞) and the pair of filtrations (G∞,F).We verify that the aforementioned pair indeed satisfies the requirements of Proposition III.7:

(i) By Definition III.29 holds G∞ ⊂ F.

III.7. STEP 1 IS VALID 101

(ii) By (M2) holds X∞ ∈ H2(G∞;R) and by construction of Y ∞ we have that Y ∞ ∈ H2(G∞;R).By Proposition III.36 we have that they are also F−martingales.(iii) The function R2,I satisfies the required properties.(iv) This is Lemma III.45.(v) Since both X∞, Y ∞ are square-integrable martingales, we apply Kunita–Watanabe Inequality I.24

in order to obtain that Var([X∞,i, Y ∞]

)∞ ∈ L1(G;R) for every i = 1, . . . , . Therefore, we can conclude

that [X∞,i, Y ∞] is of class (D) for the filtration F, for every i = 1, . . . , . Moreover, by Proposition III.36we have that [X∞,i, Y ∞]− 〈X∞,i, Y ∞〉G∞

is an F−martingale for every i = 1, . . . , .(vi) By Lemma A.6 we have that σ

(J (X∞))= B(R). Moreover, Proposition III.36 verifies that[

R2,I μ(Xd,G∞), Y ∞]− ⟨R2,I μ

(Xd,G∞), Y ∞⟩G∞∈ M(F;R), for every I ∈ J (X∞).

(vii) Since X∞, Y ∞ are both square-integrable, we have that |ΔY ∞|μX∞,d ∈ Aσ(F).

In view of the above, the claim of the proposition is true.

The following proposition justifies the title of the current subsection.

Corollary III.49. The orthogonal decomposition of Y ∞ with respect to (F, X∞,c, μX∞,d

) is given by

Y ∞ = Y0 + Z∞ ·X∞,c + U∞ μ(X∞,d,F), (III.72)

where Z∞ ∈ H2(G∞, X∞,c;R) and U∞ ∈ H2(G∞, μX∞,d

;R) are determined by the orthogonal decom-

position of Y ∞ with respect to (G∞, X∞,c, μX∞,d

); see Theorem III.3. In other words, the orthogonal

decomposition of Y ∞ with respect to (F, X∞,c, μX∞,d

) is indistinguishable from the orthogonal decompo-

sition of Y ∞ with respect to (G∞, X∞,c, μX∞,d

).

Proof. It is immediate by Lemma III.46, Lemma III.47 and Proposition III.48.

III.7. Step 1 is valid

This subsection is devoted to the proof of the main theorem. In view of the preparatory resultsobtained in the previous sections as well as of the outline of the proof presented in subsection III.2, the

following proof basically amounts to proving that N∞ = ĎN up to indistinguishability and 〈N∞〉 = Ξ. Wewill divide the proof in two main parts. Initially we will prove that the Convergence (III.3) is true, where

we will have obtained already the convergence 〈Nk〉 L−−→ 〈N∞〉. Then we will prove that the Convergence(III.4) and (III.5) are also true.

Proof of Theorem III.3. By Lemma III.26.(iii) we have that the sequence((Nk, 〈Nk〉))

k∈Nis

tight in D(R2). Therefore, an arbitrary subsequence((Nkl , 〈Nkl〉))

l∈Nhas a further subsequence indexed

by (klm)m∈N which converges in law, say to (N,Ξ), i.e.

(Nklm , 〈Nklm 〉) L−−−−→m→∞ (N,Ξ), (III.73)

where N is a cadlag process and Ξ is a continuous and increasing process. The continuity of Ξ fol-lows from Lemma III.26.(i). Therefore, we can use the results of subsection III.6 for the subsequence((Nklm , 〈Nklm 〉))

m∈Nand the pair (N,Ξ).

By Proposition III.41 and Corollary III.49 we conclude

〈Y ∞, N〉 = 〈Z∞ ·X∞,c, Nc〉+ 〈U∞ μ(X∞,d,G∞), N

d〉= 〈Z∞ ·X∞,c, N

c〉+ 〈U∞ μ(X∞,d,F), Nd〉

= Z∞ · 〈X∞,c, Nc〉+ (U∞M

μX∞,d [ΔN |PF]) ∗ ν(X∞,d,F) (III.65)

= 0, (III.74)

i.e. [Y ∞, N ] is an F−martingale. On the other hand, we have proven in Proposition III.36 that[Y ∞, N ]−Ξ is an F−martingale as well. By subtracting the two martingales we obtain that Ξ is also anF−martingale. However, Ξ is an F−predictable process of finite variation and a martingale, therefore ithas to be constant; see Corollary I.65. Now, we have that

〈Nklm 〉 L−−−−→m→∞ Ξ implies that 〈Nklm 〉0 L−−−−→

m→∞ Ξ0.


Recall that by definition 〈Nk〉0 = 0 for every k ∈ N, hence Ξ0 = 0. Therefore Ξ = 0 and, since the limitis a deterministic process, the convergence above is equivalent to the following

〈Nklm 〉 (J1(R),P)−−−−−−−−→m→∞ 0.

Since the limit above is common for every subsequence and (D, J1(R)) is Polish, we can concludefrom [28, Theorem 9.2.1] that

〈Nk〉 (J1(R),P)−−−−−−→ 0.

Using Lemma III.26.(v) and [35, Theorem 1.11] we can strengthen the above convergence to

〈Nk〉 (J1(R),L1)−−−−−−−→ 0. (III.75)

Then, we can also conclude that Nk (J1(R),L2)−−−−−−−→ 0. Indeed, for every R > 0 and by Doob’s L2−inequality

we obtain

E

[sup

t∈[0,R]

|Nkt |2]≤ 4E[|Nk

R|2] = 4E[〈Nk〉R] −−−−−−→k→∞

0,

which implies the convergence E[d2lu(Nk, 0)] −→ 0.

Using the convergence of Y k (J1(R),L2)−−−−−−−→ Y ∞ and the convergence of (Nk)k∈N to the zero process,

which is trivially continuous, we can obtain the joint convergence

(Y k, Nk)(J1(R

2),L2)−−−−−−−−→ (Y ∞, 0).

Moreover, using the orthogonal decompositions of Y k and Y ∞ and the previous results, we obtain

Zk ·Xk,c + Uk μ(Xk,d,Gk) = Y k −Nk − Y k0

(J1(R),L2)−−−−−−−→ Y ∞ − Y ∞

0 = Z∞ ·X∞,c + U∞ μ(X∞,d,G∞),

which yields then (III.3).

Thus, it is only left to prove convergence (III.5). Since the sequences (Xk)k∈Nand (Y k)k∈N

satisfy the

conditions of Theorem I.160 we obtain in particular that the sequences (Y k +Xk)k∈Nand (Y k −Xk)k∈N

also satisfy the conditions of this theorem. Therefore we can conclude that

〈Y k +Xk〉 (J1(R),P)−−−−−−→ 〈Y ∞ +X∞〉 and 〈Y k −Xk〉 (J1(R),P)−−−−−−→ 〈Y ∞ −X∞〉.By the continuity of the limiting processes, recall (M1) and [41, Theorem 4.2], using the identity

〈Y k, Xk〉 = 1

4

(〈Y k +Xk〉 − 〈Y k −Xk〉), for every k ∈ N,

and convergence (III.75), we can conclude that

(〈Y k〉, 〈Y k, Xk〉, 〈Nk〉) (J1(R×R×R),L1)−−−−−−−−−−−−−−→ (〈Y ∞〉, 〈Y ∞, X∞〉, 0).

In order to strengthen the last convergence to an L1−convergence, we only need to recall that by TheoremI.160 the sequences

(Tr[〈Xk〉∞

])k∈N

and(〈Y k〉∞

)k∈N

are uniformly integrable. Now Lemma A.2 provides

the uniform integrability of(∥∥Var(〈Y k, Xk〉)∞

∥∥1

)k∈N

, which allows us to conclude.

In view of the convergence of (Nk)k∈N to the zero process, we can also obtain the associated conver-gence for the terminal random variables.

III.8. Corollaries of the main theorem

In this section we provide two corollaries of Theorem III.3. The first one, which generalises Madan,Pistorius, and Stadje [50, Corollary 2.6] will be the useful for obtaining the stability property of BSDEs,which will be stated as Theorem IV.10. The second result deals with the special case that the sequenceof filtrations are generated by processes with independent increments. It is this property that providesus the flexibility to work on possibly different probability spaces and, thus, obtain weak approximations.

Theorem III.50. Assume the framework of Theorem III.3, where condition (M2) is substituted by thefollowing:

(M2′) For every k ∈ N,13we define ĎXk : (Xk,, Xk) ∈ H2(Gk;R)× H2,d(Gk;R) such that

13Observe that X∞ is assumed as in Theorem III.3.

III.9. COMPARISON WITH THE LITERATURE 103

(i) MµX

k,\

[∆Xk,

∣∣PGk] = 0 and

(ii) (Xk,∞ , Xk,\

∞ )L2(G;R`)−−−−−−−→k→∞

(X∞,c∞ , Xk,\∞ ).

The convergence of Theorem III.3 hold and additionally(〈Y k, Xk,〉, 〈Y k, Xk,\〉

) (J1(R2×R`×R`),L1)−−−−−−−−−−−−−−→(〈Y∞, X∞,c〉, 〈Y∞, X∞,d〉

).

Remark III.51. The Condition (M2′) is clearly stronger than (M2). So, as one may expect, we havesome better properties comparing to the framework of Theorem III.3. Let us comment briefly on twoimmediate effects.

(i) Unlike Condition (M2), under (M2′) we can approximate the continuous part of X∞,c with aconvergent sequence of martingales which may have purely discontinuous parts. This property turns outto be crucial for numerical schemes, where one needs to approximate the continuous part with discrete-time martingales, i.e. purely discontinuous martingales.

(ii) The separate convergence of the sequence of terminal random variables (Xk,∞ , Xk,\

∞ ) allows toapply Theorem III.3 for each sequence separately. This property allows in particular for the followingconvergence (

〈Y k, Xk,〉, 〈Y k, Xk,\〉) (J1(R2×R`×R`),L1)−−−−−−−−−−−−−−→

(〈Y∞, X∞,c〉, 〈Y∞, X∞,d〉

),

which will be crucial for obtaining the stability property of BSDEs.

Proof of Theorem III.50. For the following recall Lemma I.164, which verifies that 〈Xk,, Xk\〉 =0 for every k ∈ N. Now, we can apply initially the Theorem I.160 for the sequence (Xk,)k∈N in order toprove the convergence

〈Xk,〉 (J1(R`),P)−−−−−−−→ 〈X∞,c〉 (III.76)

and analogously for

〈Xk,\〉 (J1(R`),P)−−−−−−−→ 〈X∞,d〉. (III.77)

Then, using the aforementioned orthogonality, we refine the convergence of to the space of Ito generatedby Xk,

III.9. Comparison with the literature

In this section we are going to compare Theorem III.3 with the related literature, which mainlyamounts to Jacod, Meleard, and Protter [42, Theorem 3.3] and Briand, Delyon, and Memin [15, Theorem5]. In the former, Jacod et al. study the stability of (locally) square-integrable martingale representationsin the Galtchouk–Kunita–Watanabe sense. This means, that the sequence (Y k)k∈N (we use the notation

we have introduced for Theorem III.3) may consist of locally square-integrable martingales, while (Xk)k∈Nis a sequence of square-integrable martingales, which will serve as the Ito integrator. Using the analogousto our introduction scheme we could sketch [42, Theorem 3.3] as follows

Y k = Y k0 + Zk ·Xk + Nky 99K

99K

99K

Y∞ = Y∞0 + Z∞ ·X∞ + N∞

Comparing to our framework, in [42] there is no need for an analogous to (M3) condition. On the otherhand, in [42] the sequences are adapted to a single filtration. This, in particular, allows for significantsimplifications comparing to working in different stochastic bases. For example, one is allowed to calculatethe predictable quadratic variation 〈Y k, Xm〉 for k 6= m, which in general is not well-defined when thefiltration depends on the index of the sequence.

Let us discuss now about the work of Briand et al. [15]. The framework here can be regarded as aspecial case of our framework. Indeed, we retrieve the framework of [15] once we restrict X∞ to be aBrownian motion, (Xk)k∈N to be a purely discontinuous martingale that approximates Brownian motion

and, finally, assume that Gk = FXk .

CHAPTER IV

Stability of Backward Stochastic Differential Equations withJumps

This is the last chapter of the present dissertation and it deals with the stability property of BSDEs.The term stability will be used in an analogous sense as in Chapter III, i.e. if for a given convergentsequence of “inputs” we get a convergent sequence of “outputs”, then we will say that the stabilityholds. In Chapter II has been made clear what the “input” for a BSDE should be, namely a sextuple

SD := (G, T, ĎX, ξ, C, f) that satisfies Conditions (F1) - (F5) under (some) β. In other words, SD are

standard data under some β. Thus, we will say that “Backward Stochastic Differential Equations areStable”, if for an arbitrary sequence of standard data

(SDk)k∈N

which converges to the standard data

SD∞, with Υk being the associated to SDk solution of Equation (II.1) for every k ∈ N, the sequence ofsolutions (Υk)k∈N converges to the solution Υ∞. Of course, we will have to impose additional structuraland integrability conditions so that we can finally obtain the desired property, but we will be more preciselater.

The first result that dealt with the stability of BSDEs with Lipschitz generator f is [39]. In this work,the authors perturb the terminal value ξ and the integrator f , while the rest of the data are assumedfixed (in particular the driving martingale is Brownian motion), in order to obtain the required property.In other words, in [39] the approximating sequence consists of continuous semimartingales. This is notsatisfactory from the numerical point of view, where we need the approximating sequence to consist ofdiscrete-time semimartingales. To this direction, and restricting our interest in the case of a Lipschitzgenerator, Bally [5], Briand, Delyon, and Memin [14], Chevance [18], Henry-Labordere, Tan, and Touzi[36], Ma, Protter, San Martin, and Torres [49], Zhang [65] among others propose numerical schemes forapproximating BSDEs driven by Brownian motion. In the case the driving martingale is either a purelydiscontinuous martingale or a jump-diffusion, the literature is less abundant. We mention the work ofAazizi [1], of Lejay and Torres [46], which deals with the jump-diffusion whose purely discontinuousmartingale is a compensated Poisson case, of Bouchard and Elie [11], Bouchard, Elie, and Touzi [12],and of Madan et al. [50], which can be regarded as the most general among the aforementioned works.In our framework we do not distinguish between continuous-time and discrete-time approximations ofthe driving martingale, which can be a general square-integrable martingale, and, moreover, we allow forfiltration perturbation. However, we postpone a more detailed comparison with the related literature forSection IV.6.

The structure of the chapter follows. In the first section we set part of the framework by determiningthe sequence of standard data as well as the notation we are going to use in the chapter. In Section IV.1we provide the conditions the sequence of standard data should satisfy and we state the main theorem ofthe chapter. Then, in Section IV.3 we provide the main arguments for the proof of the main theorem. Itwill be evident there that the stability of martingale representation plays an important role in obtainingthe stability property of BSDEs, as we have already commented. This is not surprising, as the readermay recall that in Theorem II.14 we obtained the existence and uniqueness of the solution of a BSDE bytranslating the problem into a martingale representation one. The technical lemmata which verify ourclaims in the body of the proof are presented in Sections IV.4 and IV.5.

IV.1. Notation

The probability space (Ω,G,P) will be fixed throughout the chapter. Moreover, as we have alreadyinformed the reader in the introduction of Chapter I, for this chapter the dimension of the drivingmartingales will be equal to 1. The solution of a BSDE will be also real-valued. Moreover, we are goingto borrow the notation as was set in Section II.3. However, before we proceed, let us fix

• a sequence of filtrations (Gk)k∈N,

105

106 IV. STABILITY OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS

• a sequence of stopping times (T k)k∈N,

• a sequence (ĎXk)k∈N, where ĎXk := (Xk,, Xk,) such that (ĎXk)Tk ∈ H2(Gk;R) × H2,d(Gk;R) for

every k ∈ N and• X∞ such that (X∞)T

∞ ∈ H2(G∞;R). We denote by ĎX∞ the natural pair of X∞.

Now, we fix a sequence (SDk)k∈Nwhere SDk := (Gk, T k, ĎXk, ξk, C

ĎXk

, fk). The following assumptionwill be true for the rest of the chapter.

(SDk)k∈Nis a sequence of standard data under the same β.1

In other words, for every k ∈ N, the sextuple SDk, satisfies Conditions (F1) - (F5) of Section II.4 under

β. The reader should keep in mind that β is universal.

Notation IV.1. For every k ∈ N the Gk−stopping time T k is fixed. Therefore, in order to ease notation,

we will denote the stopped processes (ĎXk)Tk

, (Xk,)Tk

, (Xk,)Tk

, (CĎXk

)Tk

simply as ĎXk, Xk,, Xk,,

CĎXk

, for every k ∈ N. We will do so also for fk10,Tk, i.e. we will denote it simply by fk, for every

k ∈ N.

Notation IV.2. We provide here the notation which is associated to the standard data SDk, for everyk ∈ N, through the conditions (F1) - (F5).

• We adopt Notation II.2 as well as Notation II.6, since the pair (ĎXk, CĎXk

) satisfies Assumption (C)for every k ∈ N.

• For every k ∈ N, the stochastic bounds associated to the generator fk by (F3) are denoted by rk

and ϑk := (θk,c, θk,d).• For every k ∈ N, the associated processes by (F4) are denoted by αk := (max

√rk, θk,c, θk,d

)

12

and Ak :=∫(0,·](α

k)2 dCĎXk

. Moreover, the bound associated to Ak by (II.18) will be denoted by Φk, for

every k ∈ N.

In view of the above, we can adopt also Notation II.4, which provides the associated normed spaces,whose notation we immediately simplify. We provide initially the simplifications for the standard dataSDk indexed by the positive integers.

Notation IV.3. For any β ≥ 0, (ω, t) ∈ Ω× R+ and k ∈ N we define:

• Pk := PGk

, Pk := PGk

.• L2

β,k := L2Ak,β(Gk

Tk ;R).

• H2β,k := H2

Ak,β(Gk, ĎXk;R), H2,

β,k := H2β(G

k, Xk,;R), H2,β,k := H2

β(Gk, μXk,

;R).

• S2β,k := S2

Ak,β(Gk, ĎXk;R), H2,⊥

β,k := H2,⊥Ak,β

(Gk, Xk,, μXk,

;R), H2β,k := H2

Ak,β(Gk;R).

• |||·|||t,k := |||·|||ĎXk

t , Hω,t,k := HĎXk

ω,t , Hk := HĎXk

.• ‖ · ‖β,k := ‖ · ‖Ak,β,ĎXk , ‖ · ‖,β,k := ‖ · ‖,Ak,β,ĎXk .

• Ck := CĎXk

, ck := cĎXk

, μk := μXk,

, νk := ν(Xk,,Gk), μk := μ(Xk,,Gk).

The following is the analogous notation for the standard data SD∞.

Notation IV.4. For any β ≥ 0, (ω, t) ∈ Ω× R+

• P∞ := PG∞, P∞ := PG

∞.

• L2β,∞ := L2

A∞,β(G∞T∞ ;R).

• H2β,∞ := H2

A∞,β(G∞, ĎX∞;R), H2,c

β,∞ := H2β(G

∞, X∞,c;R), H2,dβ,∞ := H2

β(G∞, μX∞,d

;R).

• S2β,∞ := S2

A∞,β(G∞, ĎX∞;R), H2,⊥

β,∞ := H2,⊥A∞,β(G

∞, X∞,c, μX∞,d

;R), H2β,∞ := H2

A∞,β(G∞;R).

• |||·|||t,∞ := |||·|||ĎX∞

t , Hω,t,∞ := HĎX∞ω,t , H∞ := H

ĎX∞.

• ‖ · ‖β,∞ := ‖ · ‖A∞,β,ĎX∞ , ‖ · ‖,β,∞ := ‖ · ‖,A∞,β,ĎX∞ .

• C∞ := CĎX∞

, c∞ := cĎX∞

, μ∞ := μX∞,d

, ν∞ := ν(X∞,d,G∞), μ∞ := μ(X∞,d,G∞).

Notation IV.5. We introduce some convenient notation for two subsets of R+×R as well as the notationfor sets of continuous functions with compact support. D is an arbitrary subset of R.

• E0 := (R+ × 0) ∪ (0 × R) and E := (R+ × R) \ E0.

1We can assume that β is sufficiently large. In other words, we can assume that β is such that the values MΦ(β),

MΦ (β) determined in Lemma II.13 are arbitrarily close to 18eΦ.

IV.2. FRAMEWORK AND STATEMENT OF THE MAIN THEOREM 107

• Let f : (D, | · |) → (R, | · |), its support is the set supp(f) :=s ∈ R+, f(s) = 0

|·| ∩D.

• Let f : (R+ × R, ‖ · ‖2) → (R, | · |), its support is the set supp(f) :=s ∈ R+ × R, f(s) = 0

‖·‖2

.

• Cc(R+;R) :=Z :(R+, | · |

)→ (R, | · |), Z is continuous with compact support.

• Cc,0(R;R) :=U :(R, | · |)→ (R, | · |), U is continuous with compact support supp(U)

and such that infδR(x, 0), x ∈ supp(U) > 0.

• Cc,0(R+ × R;R) :=U :(R+ × R, ‖ · ‖2) → (R, | · |), U is continuous with compact

support supp(U) and such that inf‖x− y‖2, x ∈ supp(U), y ∈ E0

> 0.

Remark IV.6. The set Cc,0(R+ ×R;R) is non-empty. For example, let U : R+ ×R → R whose support

supp(U) is a compact subset of the open set E. Then U lies in Cc,0(R+ ×R;R), since E0 = (R+ ×R) \ Eis a closed subset of R+ × R and supp(U) ∩ E0 = ∅. In particular, U(t, ·) ∈ Cc(R;R) for every t ∈ R+.

Remark IV.7. Let μX be the integer-valued random measure associated to the cadlag process X. The

way that the stochastic integral with respect to μX has been defined is such that μX(E0) = 0. In

other words, μX(R+ × R) = μX(E). Therefore, for a filtration G such that ν(X,G) is well-defined, it

holds ν(X,G)(E0) = 0; this is a consequence of Theorem I.79. Moreover, this property will be used inSubsection IV.5.1 in order to attain the Lusin approximation we will make use of.

We have introduced the Notation IV.1 - IV.5 in order to ease notation. However, we will use in-terchangeably the complete notation when we want to underline a property. For example we will do sowhen we set the framework of the chapter, which is described by the conditions (B1) - (B9). The readerwill realise that the conditions (B1) - (B5) are nothing more than the framework of Theorem III.50,which is not surprising in view of the comments presented at the beginning of the chapter. The remain-ing conditions are such that we can obtain a convergent sequence of Lebesgue–Stieltjes integrals. Weneed only to comment beforehand on Condition (B6), which states that the sequence (Ak)k∈N

has tobe bounded. This condition implies the equivalence of the norms in the weighted spaces indexed by βwith the respective whose norm does not depend on A, i.e. β = 0. However, by Condition (B6) itis not necessarily implied that the sequence (Ck)k∈N

is bounded. We will assume, however, that Ck is

P−almost surely finite, for every k ∈ N, which is clearly weaker from the above statement. We providemore comments on the nature of the Conditions (B1) - (B9) in Subsection IV.5.6.

IV.2. Framework and statement of the main theorem

The sequence of standard data (SDk)k∈Nsatisfies the following conditions:

(B1) The filtration G∞ is quasi-left-continuous and the process ĎX∞ is G∞−quasi-left-continuous..(B2) The process ĎXk = (Xk,, Xk) ∈ H2(Gk;R)× H2,d(Gk;R), for every k ∈ N. Moreover,

(Xk,∞ , Xk,

∞ )L2(G;R)−−−−−→ (X∞,

∞ , X∞,∞ ).

(B3) The martingale X∞ possesses the G∞-Predictable Representation Property.

(B4) The filtrations converge weakly, i.e. Gk w−−→ G∞.

(B5) The random variable ξk ∈ L2(GkTk ;R) for every k ∈ N. Moreover, ξk

L2(G;R)−−−−−−−→ ξ∞.

(B6) The sequence (Ak∞)k∈N

is bounded.(B7) The generators possess additionally the following properties:

(i) For every (y, z, U) ∈ R × R × Cc,0(R) and for every k ∈ N the process(fk(t, y, z, U(t, ·))

t∈R+is

cadlag P-almost surely.(ii) For K1, resp. K2, compact subset of R, resp. R, and U ∈ Cc,0(R+ × R;R), if (λk)k∈N ⊂ Λ2 such

that

supt∈R+

|λk(t)− t| −−−−→k→∞

0

2Recall Definition I.108.


then P−almost surely

supt∈[0,N ]

∣∣f∞(λk(t), y, z, U(t, ·))− f∞

(t, y, z, U(t, ·)

)∣∣, z ∈ K2

−−−−→k→∞

0 for every N ∈ N.

(iii) For K1, resp. K2, compact subset of R, resp. R, and U ∈ Cc,0(R+ × R;R) it holds

supt∈R+

∣∣fk(t, y, z, U(t, ·))− f(t, y, z, U(t, ·))∣∣, (y, z) ∈ K1 ×K2

−−−−→k→∞

0 P− almost surely.

(iv) The sequences of stochastic bounds (rk)k∈N, (ϑk)k∈N are P-almost surely bounded on R+. Addi-tionally,

rk‖·‖∞−−−−−→ r∞, θk,c

‖·‖∞−−−−−→ θ∞,c, θk,d‖·‖∞−−−−−→ θ∞,d P-almost surely

and α∞(ω) is dC∞(ω)-almost everywhere continuous for P−almost every ω. Moreover, d〈X∞,c〉s/

dC∞shas locally bounded paths P−almost surely.

(B8) The sequence

(∫(0,∞)

eβAkt

∣∣fk(t, 0, 0,0)∣∣2

(αkt )2dCkt

)k∈N


(B9) The sequences (Ck)k∈N and (Φk)k∈N satisfy

(i) Ck(J1(R),P)−−−−−−→ C∞,

(ii) Ck∞ := limt→∞ Ckt ∈ R+ P− a.s. with Ck∞P−→ C∞∞ and

(iii) Φk|·|−−→ Φ∞ := 0.

Remark IV.8. The convergence in Condition (B5) implies that supk∈N∥∥ξk∥∥L2(G;R)

<∞. In view of the

Condition (B6) the above bound is equivalent to supk∈N∥∥ξk∥∥L2

Ak,β(GkTk

;R)<∞. Moreover, in view of the

Condition (B6) again, the Condition (B8) is equivalent to(∫(0,∞)

∣∣fk(t, 0, 0,0)∣∣2

(αkt )2dCkt

)k∈N


In the proofs of the lemmata we will need in the next sections we will always indicate the process Ak

so that it is clear its role. Moreover, this inconvenience may turn out to be fruitful in the case it helpsthe reader to provide any suggestions for the improvement of the Condition (B6). These suggestions aremore than welcome by the author.

Remark IV.9. Recall Remark II.15 and that SDk is standard data under β, for every k ∈ N. In viewof Condition (B9).(iii) we have that the equation

Yt = ξk+

∫(t,Tk]

fk(s, Ys, Zs, U(s, ·)) dCks −∫ Tk

t

Zs dXk,s −

∫ Tk

t

∫RU(s, x) µk(ds,dx)−

∫ Tk

t

dNks (IV.1)

has a unique solution (Y k, Zk, Uk, Nk) finally for k ∈ N. Without loss of generality, possibly by restricting

ourselves in the case supk∈N Φk < (18e)−1 and for β large enough (recall for this Foonote 1), we will assumethat the BSDE (IV.1) admits a unique solution for every k ∈ N. Moreover, observe that for the standarddata SD∞ we have assumed Φ∞ = 0, therefore there exists a unique solution of the BSDE

Yt = ξ∞ +

∫(t,T∞]

f∞(s, Ys, Zs, U(s, ·)) dC∞s −∫ T∞

t

Zs dX∞,cs −∫ T∞

t

∫RU(s, x) µ∞(ds,dx). (IV.2)

Observe that because of Condition (B3) the orthogonal martingale N∞ vanishes.

We proceed now to the statement of the theorem of the chapter. In the body of its proof we willpresent the main arguments, which are based on the Moore–Osgood Theorem, see Theorem A.8 and thereferences therein, while the remaining technical parts will be presented in a series of sections.

Theorem IV.10. Let the Conditions (B1) - (B9) hold for the sequence (SDk)k∈N of standard data

under the same β. For every k ∈ N denote by Υk := (Y k, Zk, Uk, Nk) the unique solution of the BSDE

(IV.1) associated to the standard data SDk and by Υ∞ := (Y∞, Z∞, U∞, 0) the associated to SD∞solution. Then,

(Y k, Zk ·Xk, + Uk ? µk, Nk)(J1(R3),L2)−−−−−−−−→ (Y∞, Z∞ ·X∞,c + U∞ ? µ∞, 0

)(IV.3)

IV.3. PROOF OF THEOREM IV.7 109

and

(〈Y k〉, 〈Zk ·Xk,〉, 〈Uk ? µk〉, 〈Y k, Xk,〉, 〈Y k, Xk,\〉)(J1(R5),L1)−−−−−−−−→

(〈Y∞〉, 〈Z∞ ·X∞,c〉, 〈U∞ ? µ∞〉, 〈Y∞, Xc〉, 〈Y∞, Xd〉).(IV.5)

IV.3. Proof of Theorem IV.7

For the rest of the chapter Conditions (B1) - (B9) will be assumed to be in force for the sequence

of (SDk)k∈N. Before we proceed to the proof of Theorem IV.10 we present the next result, which givesuniform a priori estimates of the tail of the Picard approximations, see Corollary II.22. Proposition IV.11generalizes Briand et al. [15, Corollary 10].3 In the following, for k ∈ N, we will keep the meaning of(Υk)k∈N as defined in Theorem IV.10.

Proposition IV.11. There exist k0, k?,0 ∈ N such that

limp→∞

supk≥k0

∥∥Υk −Υk,(p)∥∥2

ˆβ,k

= 0 and limp→∞

supk≥k?,0

∥∥Υk −Υk,(p)∥∥2

?,ˆβ,k

= 0,

where Υk,(p) is the p-element of the Picard approximation of the solution Υk associated to the standarddata SDk.

Proof. Let us initially restate for the convenience of the reader the constants ΠΦ(γ, δ), ΠΦ? (γ, δ) as

well as the values MΦ(β), MΦ? (β) of Lemma II.16, where β > 0 and γ, δ ∈ (0, β] such that γ < δ. So,

from Lemma II.16, we have

ΠΦ(γ, δ) :=9

δ+ (2 + 9δ)

e(δ−γ)Φ

γ(δ − γ)

with

MΦ(β) := inf(γ,δ)∈Cβ

ΠΦ(γ, δ) =9

β+

Φ2(2 + 9β)√β2φ2 + 4− 2

exp(βΦ + 2−

√β2Φ2 + 4

2

)and

ΠΦ? (γ, δ) :=

8

γ+

9

δ+ 9δ

e(δ−γ)Φ

γ(δ − γ)with MΦ

? (β) := inf(γ,δ)∈Cβ

ΠΦ? (γ, δ) = ΠΦ

? (γΦ? (β), β).

We observe that, when we fix γ, δ ∈ (0,ˆβ]2, then the functions

R+ 3 x 7→ Πx(γ, δ) ∈ R+ and R+ 3 x 7→ Πx?(γ, δ) ∈ R+ (IV.6)

are decreasing and continuous under the usual topology of R. Hence, by the assumption (B9).(iii) andby Footnote 1 on p. 106, we can assume that there exist Φk0 and Φk?,0 such that

MΦk0(β) = ΠΦk0

(γΦk0, β) < 1

2 , for some γΦk0 ∈ (0,ˆβ]2 and

MΦk?,0? (β) = ΠΦk?,0

? (γΦk?,0? , β) < 1

2 , for some γΦk?,0 ∈ (0,ˆβ]2.

Consequently, by Lemma II.13 and the fact that the functions in (IV.6) are decreasing, we obtain

ΠΦk(γΦk , β) = MΦk(β) ≤ ΠΦk(γΦk0, β) ≤MΦk0

(β) <1

2for every k ≥ k0,

as well as

ΠΦk

? (γΦk

? , β) = MΦk

? (β) ≤ ΠΦk

? (γΦk?,0? , β) ≤MΦk?,0

? (β) <1

2for k ≥ k?,0. (IV.7)

Let us, now, for the rest of the proof to deal only with the second convergence, since we can readily adaptthe same arguments for the first one. By Corollary II.22, more precisely by (II.53), we derive

‖Υk −Υk,(p)‖2?,

ˆβ,k≤∞∑n=0

2n+1‖Υk,(p+n) −Υk,(p+n+1)‖2?,

ˆβ,k

≤∞∑n=0

2n+12−(p+n)‖Υk,(1)‖2?,

ˆβ,k

= 41−p‖Υk,(1)‖2?,

ˆβ,k. (IV.8)

3For more details, the interested reader should consult Section IV.6, where we will provide the comparison with therelated literature.


Choosing in the Picard approximation our starting point to be the zero element for every k ∈ N, weobtain by Lemma II.16∥∥Υk,(1)

∥∥2

?,β,k≤ Πβ,Φk

?

∥∥ξk∥∥2

L2β,k

+ ΠΦk

? (γΦk

? , β)

∥∥∥∥fk(·, 0, 0,0)

αk

∥∥∥∥2

H2β,k

.

In view of Condition (B5), Condition (B8), Remark IV.8 and the fact that the function R+ 3 x 7→Πβ,x? ∈ R+ (resp. R+ 3 x 7→ Πβ,x ∈ R+) is decreasing, we can obtain with the help of (IV.7)

supk≥k?0

‖Υk,(1)‖2?,

ˆβ,k≤ Πβ,Φk?,0 sup

k≥k?0‖ξk‖2L2

ˆβ

(GkTk

) + ΠΦk

? (γΦk?,0? , β) sup

k≥k?0

∥∥∥∥fk(·, 0, 0,0)

αk

∥∥∥∥2

H2?,β,k

<∞, (IV.9)

which implies the desired result in conjunction with Inequality (IV.8).

IV.3.1. Main part of the proof of Theorem IV.10. As we have already commented, we willprovide below the main arguments for proving Theorem IV.10. The technical results which verify ourargumentation will be provided in the coming sections.

Proof of Theorem IV.10. We will keep the notation of Proposition IV.11 and we will denote by(Υ∞,(p))p∈N the Picard scheme associated to SD∞. The starting point of the Picard scheme associated

to the standard data SDk will be the zero element, for every k ∈ N. In other words, we define Υk,(0) :=(0, 0,0, 0), for every k ∈ N. Recall that for every k ∈ N we have stopped the processes indexed by k attime T k. Therefore, we can substitute for every k ∈ N the terminal time T k by ∞ and we will do sofor our convenience. Then, by the construction of Picard scheme (recall Corollary II.22) holds for everyp ∈ N ∪ 0

Y∞,(p+1)t = ξ∞ +

∫(t,T∞]

f∞(s, Y∞,(p)s , Z∞,(p)s , U∞,(p)(s, ·)

)dC∞s

−∫ T∞

t

Z∞,(p+1)s dX∞,cs −

∫ T∞

t

∫RU∞,(p+1)(s, x)µ∞(ds,dx)

= ξ∞ +

∫(t,∞)

f∞(s, Y∞,(p)s , Z∞,(p)s , U∞,(p)(s, ·)

)dC∞s

−∫ ∞t

Z∞,(p+1)s dX∞,cs −

∫ ∞t


(IV.11)

Analogously, for every k ∈ N and for every p ∈ N ∪ 0

Yk,(p+1)t = ξk +

∫(t,Tk]

fk(s, Y k,(p)s , Zk,(p)s , Uk,(p)(s, ·)

)dCks

−∫ Tk

t

Zk,(p+1)s dXk,

s −∫ Tk

t

∫RUk,(p+1)(s, x) µk(ds,dx)−

∫ Tk

t

dNk,(p+1)s

= ξk +

∫(t,∞)


)dCks

−∫ ∞t

Zk,(p+1)s dXk,

s −∫ ∞t


∫ ∞t

dNk,(p+1)s

(IV.12)

By Proposition IV.11 we get that Υk,(p)‖·‖β,k−−−−→p→∞

Υk as well as Υk,(p)‖·‖?,β,k−−−−−−→p→∞

Υk, both finally uniformly

for k ∈ N. In particular, the latter convergence implies Y k,(p)S2β,k−−−→

p→∞Y k finally uniformly in k, which in

turn implies Y k,(p)(J1(R),L2)−−−−−−−→

p→∞Y k finally uniformly in k. Hence, by orthogonality of the respective parts,

Ito’s isometry and Doob’s inequality (see also the proof of Theorem II.14 on p. 68) we obtain

(Y k,(p), Zk,(p) ·Xk,, Uk,(p) ? µk, Nk,(p))(‖·‖∞,L2)−−−−−−−→p→∞

(Y k, Zk ·Xk,, Uk ? µk, Nk)

finally uniformly in k. At this point we will combine two facts in order to rewrite the above convergenceunder the J1−topology. The first one is that the convergence under ‖ · ‖∞−norm allows us to conclude

IV.3. PROOF OF THEOREM IV.7 111

the convergence of the sum of two convergent sequences. The second is that the a ‖ · ‖∞−convergentsequence is also J1−convergent. Therefore,

(Y k,(p), Zk,(p) ·Xk, + Uk,(p) ? µk, Nk,(p))(J1(R3),L2)−−−−−−−−→p→∞

(Y k, Zk ·Xk, + Uk ? µk, Nk) (IV.13)

finally uniformly in k. Consequently, in order to apply the Moore-Osgood Theorem, see Theorem A.8, itis sufficient to prove that for every p ∈ N ∪ 0 holds

(Y k,(p), Zk,(p) ·Xk, + Uk,(p) ? µk, Nk,(p))(J1(R3),L2)−−−−−−−−→k→∞

(Y∞,(p), Z∞,(p) ·X∞,c + U∞,(p) ? µ∞, N∞,(p)).(IV.14)

To this end, let us transform the BSDE (IV.11) and BSDE (IV.12) into martingales.4 The aforementionedtransformation will allow us to utilise the stability of martingale representations. For fixed p ∈ N ∪ 0we have

M∞,(p)t := Y

∞,(p+1)t +

∫(0,t]

f∞(s, Y∞,(p)s , Z∞,(p)s , U∞,(p)(s, ·)

)dC∞s (IV.15)

(IV.11)= ξ∞ +

∫(0,∞)

f∞(s, Y∞,(p)s , Z∞,(p)s , U∞,(p)(s, ·)

)dC∞s

−∫ ∞t

Z∞,(p+1)s dX∞,cs −

∫ ∞t


(IV.11)= Y

∞,(p+1)0 +

∫ ∞0

Z∞,(p+1)s dX∞,cs +

∫ ∞0


−∫ ∞t

Z∞,(p+1)s dX∞,cs −

∫ ∞t


= Y∞,(p+1)0 +

∫ t

0

Z∞,(p+1)s dX∞,cs +

∫ t

0

∫RU∞,(p+1)(s, x)µ∞(ds,dx) (IV.16)

is a square integrable G∞-martingale on R+. Analogously, for k ∈ N and p ∈ N ∪ 0 we have

Mk,(p)t := Y

k,(p+1)t +

∫(0,t]


)dCks (IV.17)

(IV.12)= ξk +

∫(0,∞)


)dCks

−∫ ∞t

Zk,(p+1)s dXk,

s −∫ ∞t


∫ ∞t

dNk,(p+1)s

(IV.12)= Y

k,(p+1)0 +

∫ ∞0

Zk,(p+1)s dXk,

s +

∫ ∞0

∫RUk,(p+1)(s, x) µk(ds,dx) +

∫ ∞0

dNk,(p+1)s

−∫ ∞t

Zk,(p+1)s dXk,

s −∫ ∞t


∫ ∞t

dNk,(p+1)s

= Yk,(p+1)0 +

∫ t

0

Zk,(p+1)s dXk,

s +

∫ t

0

∫RUk,(p+1)(s, x) µk(ds,dx) +

∫ t

0

dNk,(p+1)s (IV.18)

is a square integrable Gk-martingale on R+. At this point, we can obtain the convergence

(Zk,(p+1) ·Xk, + Uk,(p+1) ? µk, Nk,(p+1))(J1(R3),L2)−−−−−−−−→

k→∞(Z∞,(p+1) ·X∞,c + U∞,(p+1) ? µ∞, 0) (IV.19)

if we apply Theorem III.50 to the sequence (Mk,(p))k∈N. In view of the Conditions (B1) - (B4), we needonly to prove the convergence

Mk,(p)∞

L2(G;R)−−−−−→k→∞

M∞,(p)∞ (IV.20)

for every p ∈ N∪ 0. However, in view of (IV.15) and (IV.17) and recalling that for the Picard schemes

holds Yk,(p+1)∞ = ξk, for every k ∈ N and p ∈ N∪0, we obtain immediately a pair of sufficient conditions

for the Convergence (IV.20) which read

4Recall that the same technique was used in the proof of Theorem II.14 in order to utilise finally the orthogonaldecomposition of square-integrable martingales.


ξkL2(G;R)−−−−−→ ξ∞; but this is Condition (B5).

∫(0,∞)


)dCks

L2(G;R)−−−−−→k→∞

∫(0,∞)

f∞(s, Y∞,(p)s , Z∞,(p)s , U∞,(p)(s, ·)

)dC∞s for

every p ∈ N ∪ 0. We will name for later reference the one which corresponds to the Lebesgue-Stieltjesintegrals by∫

(0,∞)


)dCks

L2(G;R)−−−−−→k→∞∫(0,∞)

f∞(s, Y∞,(p)s , Z∞,(p)s , U∞,(p)(s, ·)

)dC∞s .

(LS(p)∞ )

Assume for the following that (LS(p)∞ ) is valid for every p ∈ N∪0, i.e. we can apply Theorem III.50 for

the martingale sequence (Mk,(p))k∈N. Therefore, we obtain the convergence for every p ∈ N ∪ 0

(Mk,(p), Zk,(p+1) ·Xk, + Uk,(p+1) ? µk, Nk,(p+1))(J1(R3),L2)−−−−−−−−→

(M∞,(p), Z∞,(p+1) ·X∞,c + U∞,(p+1) ? µ∞, 0)(IV.21)

Comparing now the above convergence with Convergence (IV.14), we realise that they differ only on thefirst element. Recall once again the definition of Mk,(p), for k ∈ N; see (IV.15) and (IV.17). Now, we canobtain the convergence

(Mk,(p), Y k,(p+1))(J1(R2),L2)−−−−−−−−→ (M∞,(p), Y∞,(p+1)) for every p ∈ N ∪ 0,

if the convergence∫(0,·]


)dCks

(J1(R),L2)−−−−−−−→∫

(0,·]f∞(s, Y∞,(p)s , Z∞,(p)s , U∞,(p)(s, ·)

)dC∞s .

(LS(p))

holds for every p ∈ N ∪ 0. The last argument is indeed valid in view of the continuity of the limitprocess (recall (B9).(iii) and in particular that Φ∞ = 0), Proposition I.120 and Corollary I.122.

To sum up, in order to obtain the convergence (IV.14) for every p ∈ N ∪ 0 we need to prove the

Convergence (LS(p)) as well as the Convergence (LS(p)∞ ).

Let us comment now on the way that we will obtain the Convergence (IV.5). Initially recall the formof the sequence (Y k,(p))k∈N for every p ∈ N ∪ 0. By (IV.15) and (IV.17) we have for p ∈ N ∪ 0

Y k,(p+1) = Mk,(p) −∫

(0,·]fk(s, Y k,(p)s , Zk,(p)s , Uk,(p)(s, ·)

)dC∞s

where Y k,(0) is the zero process k ∈ N and Mk,(0) is determined by the triplet(Zk,(0), Uk,(0), Nk,(0)

)for

every k ∈ N. Therefore, for p ∈ N, Y k,(p) is a Gk−semimartingale, for every k ∈ N, but not necessarily inSsp(Gk;R) because of the presence of the cadlag and Gk−adapted process Lebesgue–Stieltjes integral. In

particular, for (Mk,(p))k∈N we conclude that it is P-UT by Lemma I.143 as J1(R)−convergent martingale

sequence and L2−bounded; we have assumed the validity of (IV.20). For the sequence (for fixed p ∈N ∪ 0) of the Lebesgue–Stieltjes integrals we can conclude it is also P-UT by Proposition I.151 andthe convergence (LS(p)). Therefore, (Y k,(p))k∈N is P-UT, for every p ∈ N, as sum of P-UT sequences.Consequently, by Theorem I.152 we obtain the convergence[

Y k,(p)] J1(R),P−−−−−→

k→∞

[Y∞,(p)

]for every p ∈ N.

Not surprisingly, we will apply Theorem I.159 for the sequence ([Y k,(p)])k∈N, for every p ∈ N. It is

immediate by Proposition I.98 that for every k ∈ N [Y k,(p)] is a special Gk-martingale with canonicaldecomposition [Y k,(p)] = ([Y k,(p)]−〈Y k,(p)〉)+ 〈Y k,(p)〉, for every p ∈ N. Then, using standard argumentsand the polarisation identity we are able to obtain Convergence (IV.5).

It is clear by the arguments provided in the proof of Theorem IV.10 that we are going to prove by

induction on p ∈ N∪ 0 the Convergence (LS(p)) and (LS(p)∞ ), which we restate here for the convenience

IV.4. THE FIRST STEP OF THE INDUCTION IS VALID 113

of the reader:∫(0,·]


)dCks

(J1(R),L2)−−−−−−−→

k→∞∫(0,·]

f∞(s, Y∞,(p)s , Z∞,(p)s , U∞,(p)(s, ·)

)dC∞s

(LS(p))

and∫(0,∞)


)dCks

L2(G;R)−−−−−→k→∞∫(0,∞)

f∞(s, Y∞,(p)s , Z∞,(p)s , U∞,(p)(s, ·)

)dC∞s .

(LS(p)∞ )

Notation IV.12. In the next sections we will additionally use the following notation.

• Lk,(p)t :=

∫(0,t]


)dCks , for t ∈ [0,∞), k ∈ N and p ∈ N ∪ 0.

• Lk,(p)∞ :=

∫(0,∞)


)dCks , for k ∈ N and p ∈ N ∪ 0.

• Γk,(p) :=

∫(0,∞)

eβAks

∣∣fk(s, Y k,(p)s , Zk,(p)s , Uk,(p)(s, ·)

)∣∣2(αks )2

dCks , for k ∈ N and p ∈ N ∪ 0.

• ∆k,(p) := Tr[〈Zk,(p) ·Xk, + Uk,(p) ? µk〉∞

], for k ∈ N and p ∈ N ∪ 0.

• ∆∞,(p) := Tr[〈Z∞,(p) ·X∞,c + Uk,(p) ? µ∞〉∞

], p ∈ N ∪ 0.

• µCk(ω) denotes the measure on(R+,B(R+)

)with associated distribution function Ck(ω) P−almost

surely.

IV.4. The first step of the induction is valid

Recall that in the proof of Proposition IV.11 we have set(Y k,(0), Zk,(0), Uk,(0)

):= (0, 0,0) for every

k ∈ N. Now we provide some useful lemmata that we will utilise for proving the first step of the inductionin Proposition IV.16.

Lemma IV.13. The sequence (Ck)k∈N is P-UT.

Proof. Firstly recall the fact that (Ck)k∈N is a sequence of increasing processes. Therefore, Var(Ck) =

Ck, for every k ∈ N.On the other hand, Condition (B9).(i) implies that (Ck)k∈N is tight in D(R), which in

turn implies that Var(Ck)t is tight in R for every t ∈ R+. Now, we can conclude by Proposition I.151.

Lemma IV.14. For any subsequence (Ckl)l∈N there exists a further subsequence (Cklm )m∈N such that

CklmJ1(R)−−−−→m→∞

C∞ P− almost surely as well as Cklm∞

J1(R)−−−−→m→∞

C∞∞ P− almost surely.

Moreover, µCklm

w−−→ µC∞ P−almost surely.

Proof. The first statement is direct by Theorem I.141. Indeed, the fact that (D(R), δJ1) and (R, δ|·|)

are both Polish and Conditions (i), (ii) of (B9) allow us to verify the statement. Passing possibly to afurther subsequence we can assume without loss of generality that both convergent sequences are indexedby (klm)k∈N. The second statement is also true in view of Condition (B9).(iii), where the conditionΦ∞ = 0 implies that C∞ is continuous, Lemma A.13 and Proposition A.12.

Lemma IV.15. The sequence

supt∈R+

∣∣Lk,(0)t

∣∣2, k ∈ N


Proof. Using Cauchhy–Schwarz Inequality as in (II.25) we can obtain for every t ∈ [0,∞] and k ∈ N

|Lk,(0)t |2 ≤ 1

β

∫(0,t]

eβAks

∣∣fk(s, 0, 0,0)∣∣2

(αks )2dCks ≤

1

β

∫(0,∞)

eβAks

∣∣fk(s, 0, 0,0)∣∣2

(αks )2dCks .

Since, by (B8), the right-hand side is independent of t and uniformly integrable, we obtain the requiredresult by Corollary I.29.(iv).


Proposition IV.16. The first step of the induction is valid, i.e.∫(0,·]

fk(s, Y k,(0)

s , Zk,(0)s , Uk,(0)(s, ·)

)dCks

(J1(R),L2)−−−−−−−→

k→∞∫(0,·]

f∞(s, Y∞,(0)

s , Z∞,(0)s , U∞,(0)(s, ·)

)dC∞s .

(LS(0))

and∫(0,∞)

fk(s, Y k,(0)

s , Zk,(0)s , Uk,(0)(s, ·)

)dCks

L2(G;Rn)−−−−−−→k→∞∫(0,∞)

f∞(s, Y∞,(0)

s , Z∞,(0)s , U∞,(0)(s, ·)

)dC∞s .

(LS(0)∞ )

Proof. By definition(Y k,(0), Zk,(0), Uk,(0)

):= (0, 0,0) for every k ∈ N. We are going to apply

Vitali’s theorem (see Theorem I.32), i.e. we will prove initially the Convergence (LS(0)) and (LS(0)∞ )

in probability. This will be proven by means of Theorem I.141. Let a subsequence(Lkl,(0)∞

)l∈N. By

Lemma IV.14 there exists a further subsequence (klm)m∈N such that µCklm

w−−→ µC∞ P−almost surely.Consequently, we have also that supm∈N µCklm (R+) <∞ P−almost surely.

Therefore,∣∣∣∣∫(0,∞)

fklm(s, 0, 0,0

)dC

klms −

∫(0,∞)

f∞(s, 0, 0,0

)dC∞s

∣∣∣∣≤∣∣∣∣∫

(0,∞)

fklm(s, 0, 0,0

)dC

klms −

∫(0,∞)

f∞(s, 0, 0,0

)dC

klms

∣∣∣∣+

∣∣∣∣∫(0,∞)

f∞(s, 0, 0,0

)dC

klms −

∫(0,∞)

f∞(s, 0, 0,0

)dC∞s

∣∣∣∣≤∫

(0,∞)

∣∣∣fklm (s, 0, 0,0)− f∞(s, 0, 0,0)∣∣∣∞

dCklm∞

+ 2

∣∣∣∣∫(0,∞)

f∞(s, 0, 0,0

)dC

klms −

∫(0,∞)

f∞(s, 0, 0,0

)dC∞s

∣∣∣∣≤∣∣∣fklm (s, 0, 0,0)− f∞(s, 0, 0,0)∣∣∣

∞supk∈N

µCklm

(R+)

+

∣∣∣∣∫(0,∞)

f∞(s, 0, 0,0

)dC

klms −

∫(0,∞)

f∞(s, 0, 0,0

)dC∞s

∣∣∣∣ −−−−→m→∞0 P− a.s., (IV.22)

The first summand converges to zero in view of Condition (B7).(iii) and the finiteness of supm∈N µCklm (R+).The second summand converges to zero in view of the cadlag property of the paths of f∞(·, 0, 0,0) and

µCklm

w−−→ µC∞ ; use either Theorem A.14 or Mazzone [51, Theorem 1]. Therefore, the subsequence

(Lklm ,(0)∞ )m∈N converges P−a.s. to L∞,(0)

∞ . By Theorem I.141, we obtain the convergence Lk,(0)∞

P−→ L∞,(0)∞ ,

which can be strengthened to (LS(0)∞ ) in view of Lemma IV.15 and Vitali’s theorem.

We will prove now the Convergence (LS(0)). Initially we will obtain the convergence in probability.To this end, recall that by Lemma IV.13 the sequence (Ck)k∈N is P-UT. Now, by Proposition I.151, it issufficient, since C∞ is continuous, to prove the convergence of the integrands under the J1−topology inprobability. But this is immediate in view of (B7).(iii). To conclude indeed the Convergence (LS(0)) wehave only to recall Lemma IV.15 and the inequality for α, β ∈ D(Rn)

δJ1(Rn)(α, β) ≤ ‖α‖∞ + ‖β‖∞.

Although we have accomplished the aim this section, we will need to obtain some further results. Thereason is that we need to complement our induction hypothesis with the assumption that the uniformintegrability of the sequence5(∫

(0,∞)

eβAks

∥∥fk(s, Y k,(p), Zk,(p)s , Uk,(p)(s, ·))∥∥2

2

(αks )2dCks

)k∈N

5We could have abstained from writing the processes Ak, for k ∈ N, since they are bounded.

IV.4. THE FIRST STEP OF THE INDUCTION IS VALID 115

is inherited by the uniform integrability of the sequence(∫(0,∞)

eβAks

∥∥fk(s, Y k,(p−1), Zk,(p−1)s , Uk,(p−1)(s, ·)

)∥∥2

2

(αks )2dCks

)k∈N

and that of the sequences

(∣∣ξk∣∣2)k∈N and

(∫(0,∞)

eβAks

∥∥fk(s, 0, 0,0)∥∥2

2

(αks )2dCks

)k∈N

.

Corollary IV.17. The convergence(Y k,(1), Zk,(1) ·Xk, + Uk,(1) ? µk, Nk,(1)

) (J1(R3),L2)−−−−−−−−→k→∞

(Y∞,(1), Z∞,(1) ·X∞,c + U∞,(1) ? µ∞, 0

)and (

〈Zk,(1) ·Xk,〉, 〈Uk,(1) ? µk〉, 〈Nk,(1)〉) (J1(R3),L2)−−−−−−−−→

k→∞

(〈Z∞,(1) ·X∞,c〉, 〈U∞,(1) ? µ∞〉, 0

)are valid.

Proof. In view of the discussion in the body of the main theorem, the validity of LS(0) and LS(0)∞

allows us to apply Theorem III.50 and obtain (IV.21) for p = 0 and the second convergence of the state-ment above. However, the subsequent to (IV.21) comments allow us to obtain also the first convergenceof the theorem.

Remark IV.18. Observe that Y k,(1)∞ , Y (1)

∞ are well-defined, for every k ∈ N, P-almost surely. Indeed, by(IV.15) and (IV.17)

Y∞,(1)t = M

∞,(0)t −

∫ t

0

f∞(s, Y∞,(0)

s , Z∞,(0)s , U∞,(0)(s, ·)

)) dC∞s

and

Yk,(1)t = M

k,(0)t −

∫(0,t]

fk(s, Y k,(0)

s , Zk,(0)s , Uk,(0)(s, ·)

)dCks for every k ∈ N.

The limit at ∞ of the martingale part exists due to their square integrability and the limit of theLebesgue-Stieltjes integrals exists P-almost surely.

The following results will allow us to conclude the uniform integrability of the families∣∣Lk,(0)

t

∣∣2, t ∈[0,∞) and k ∈ N

and

(Tr[〈Mk,(0)〉∞

])k∈N. The reader may recall Notation IV.12 on p. 113.

Lemma IV.19. The sequence(

supt∈R+

∣∣Mk,(0)t −Mk,(0)

0

∣∣2)k∈N is uniformly integrable. In particular,

the sequence (∆k,(1))k∈N is uniformly integrable.

Proof. Let us start with an observation. The martingale Mk,(0), for k ∈ N, was defined in (IV.15)for k =∞ and in (IV.17) for k ∈ N. However, we will use the equivalent forms (IV.16) and (IV.18). Now,we proceed to prove the claim of the lemma. We will make use of the pathwise estimates in Lemma II.18and in particular (II.39). In view of (B6) and of Corollary I.29.(i) we can substitute the constants thatappear by one independent of k, say B. We have, now, for every k ∈ N that

supt∈R+

∣∣Mk,(0)t −Mk,(0)

0

∣∣2≤ C sup

t∈R+

E[∣∣ξk∣∣2 +

∫(0,∞)

eβAks

∣∣fk(s, 0, 0,0)∣∣2(αks )2

dCks

∣∣∣∣Gt]+ C

∫(0,∞)

eβAks

∣∣fk(s, 0, 0,0)∣∣2(αks )2

dCks .

At this point, we observe that the right hand side consists of elements of uniformly integrable sequence.

Let us argue a bit more on this. By (B5) and Vitali’s Theorem we have that the sequence(∣∣ξk∣∣2)

k∈N is

uniformly integrable and by (B8) we have that the sequence(∫(0,∞)

eβAks

∣∣fk(s, 0, 0,0)∣∣2(αks )2

dCks

)k∈N

.


Observe that the elements of the latter sequence appear in the above bound twice. By Corollary I.29.(iv),it is therefore sufficient for the validity of our claim to prove that the sequence(

supt∈R+

E

[∣∣ξk∣∣2 + ∫(0,∞)

eβAks

∣∣fk(s, 0, 0,0

)∣∣2(αk

s )2

dCks

∣∣∣∣Gkt

])k∈N

(IV.23)

is uniformly integrable. But this is Lemma I.106 in conjunction with Corollary I.29.(iv).We proceed now to prove that the second sequence is uniformly integrable. We will prove initially

that the sequence(〈Mk,(0) − M

k,(0)0 〉∞

)k∈N

is uniformly integrable. By the previous step and by de LaVallee Poussin–Meyer Criterion there exists Υ ∈ YFmod such that

supk∈N

E

[Υ(supt∈R+

∣∣Mk,(0)t −M

k,(0)0

∣∣2)] < ∞. (IV.24)

We apply now Theorem I.55 for every k ∈ N for the Gk−submartingale∣∣Mk,(0)

· −Mk,(0)0

∣∣2 and we obtain

E

[Υ(〈Mk,(0) −M

k,(0)0 〉∞

)] ≤ (2scΥ)scΥE

[Υ(supt∈R+

∣∣Mk,(0)t −M

k,(0)0

∣∣2)],which, in view of (IV.24) and the de La Vallee Poussin–Meyer Criterion, yields the uniform integrability

of the sequence Tr[⟨Mk,(0) −M

k,(0)0

⟩∞].

Now, in order to prove the uniform integrability of (Δk,(1)) we have initially to use the fact that

Nk,(1) ∈ H2,⊥(Gk, Xk,, μXk,

;Rn) for every k ∈ N and N∞,(1) ∈ H2,⊥(G∞, X∞,c, μX∞,d

;Rn). Then thedomination

0 ≤ Δk,(1) ≤ Δk,(1) + 〈Nk,(1)〉∞ = 〈Mk,(0)〉∞and the uniform integrability of the sequence consisting of the dominant elements of the above inequalityimply the required result.

Lemma IV.20. The sequence (Γk,(1))k∈Nis uniformly integrable.

Proof. We will prove the pathwise estimates we have obtained in Lemma II.18 and more preciselythe Estimate (II.37). As we did in the previous lemma, in view of (B6) and of Corollary I.29.(i), wewill substitute the coefficients that appear on the right-hand side of the Estimate (II.37) by a constantindependent of k, say C. Let, now, k ∈ N. By definition of Γk,(1) we have

Γk,(1) =

∫(0,∞)

eβAks

∣∣fk(s, Y

k,(1)s , Z

k,(1)s , Uk,(1)(s, ·))∣∣2(αk

s )2

dCks

(II.16)

≤(II.49)

∫(0,∞)

eβAks

[(αk

s )2∣∣Y k,(1)

s

∣∣2 + ∣∣cksZk,(1)s

∣∣2 + ∣∣∣∣∣∣Uk,(1)s (·)∣∣∣∣∣∣] dCk

s +

∫(0,∞)

eβAks

∣∣fk(s, 0, 0,0

)∣∣2(αk

s )2

dCks

(II.17)=

Lem. II.5

∫(0,∞)

eβAks

∣∣Y k,(1)s

∣∣2 dAks +

∫(0,∞)

eβAks d〈Zk,(1) ·Xk, + Uk,(1) μk〉s

+

∫(0,∞)

eβAks

∣∣fk(s, 0, 0,0

)∣∣2(αk

s )2

dCks

(II.17)

≤Lem. II.18

C supt∈R+

E

[∣∣ξ∣∣2∣∣∣Gkt

]+ C sup

t∈R+

E

[ ∫(0,∞)

eβAks

∣∣fk(s, 0, 0,0

)∣∣2(αk

s )2

dCks

∣∣∣∣Gkt

]+ eβA

k∞〈Zk,(1) ·Xk, + Uk,(1) μk〉∞ +

∫(0,∞)

eβAks

∣∣fk(s, 0, 0,0

)∣∣2(αk

s )2

dCks

The analogous bound can be proven to hold also for k = ∞. We observe again at this point that theelements of the right hand side forms a uniformly integrable sequence. For the sequence associated tothe first two summands we use analogous arguments to (IV.23). For the sequence associated to thethird summand we use Condition (B6) and Lemma IV.19. Finally, the sequence associated to the lastsummand is uniformly integrable by (B8). Then, Corollary I.29.(iv) allows us to conclude.

Proposition IV.21. The sequencesupt∈R+

∣∣Lk,(1)t

∣∣2, k ∈ Nis uniformly integrable.

IV.5. THE P−STEP OF THE INDUCTION IS VALID 117

Proof. Using the Cauchy–Schwarz Inequality as in (II.25) we can obtain for every t ∈ [0,∞] andk ∈ N

|Lk,(1)t |2 ≤ 1

β

∫(0,t]

eβAks

∣∣fk(s, Y k,(1), Zk,(1), Uk,(1)(t, ·))∣∣2

(αks )2dCks

≤ 1

β

∫(0,∞)

eβAks

∣∣fk(s, Y k,(1), Zk,(1), Uk,(1)(t, ·))∣∣2

(αks )2dCks .

Since, the right-hand side is independent of t and uniformly integrable by Lemma IV.20, we obtain therequired result by Corollary I.29.(iv).

Remark IV.22. The validity of Lemma IV.20 allows us indeed to complement our induction step withthe statement

the sequence(Γk,(p))k∈N is uniformly integrable (Γ(p)−UI)

which will allow us to prove that the sequence

supt∈R+

∣∣Lk,(p)t

∣∣2, k ∈ N


IV.5. The p−step of the induction is valid

In this section we assume that the Convergence

Lk,(p−1) (J1(R),L2)−−−−−−−→

k→∞L∞,(p−1) (LS(p−1))

and

Lk,(p−1)∞

L2(G;R)−−−−−→k→∞

L∞,(p−1)∞ (LS

(p−1)∞ )

as well as the statement

the sequences(Γk,(p−1)

)k∈N and

(∆k,(p−1)

)k∈N are uniformly integrable (Step (p− 1) UI)

are true for some arbitrary but fixed p ∈ N. We will prove the Convergence (LS(p)) and (LS(p)∞ ) as well

as the statement

the sequences(Γk,(p))k∈N and

(∆k,(p)

)k∈N are uniformly integrable (Step (p) UI)

are true.Comparing to the first step of the induction the p-step is more involved, so let us articulate in

the way we are going to reduce the complexity. In view of Vitali’s Theorem, it is sufficient to prove

initially that the Convergence (LS(p)) and (LS(p)∞ ) holds in probability and then we have to prove that the

sequences are sufficiently uniformly integrable. Now, in order to obtain the aforementioned convergencein probability, we are going to use that

(D(R), δJ1(R)

)and

(R, | · |

)are Polish spaces. Therefore, in view

of Theorem I.141, it is sufficient to prove that from every subsequence(Lkl,(p)

)l∈N, resp.

(Lkl,(p)∞

)l∈N, we

can extract a further subsequence(Lklm ,(p)

)m∈N, resp.

(Lklm ,(p)∞

)m∈N, such that

Lklm ,(p)J1(R)−−−−→m→∞

L∞,(p) P− a.s. (IV.25)

resp.

Lklm ,(p)∞

|·|−−−−→m→∞

L∞,(p)∞ P− a.s.. (IV.26)

To this end let us fix an ε > 0 and assume that for P−almost every ω there exist suitable(Zε,(p)(ω, s)

)s∈R+

and(Uε,(p)(ω, s, x)

)(s,x)∈R+×R`

such that∣∣∣∣∫(0,∞)

f∞(, s, Y∞,(p)s , Zε,(p)s , Uε,(p)(s, ·)

)dC∞s − L∞,(p)∞

∣∣∣∣ < ε

4P− a.s.. (IV.27)

Then, using the set inclusions[∣∣∣ 4∑i=1

Xk∣∣∣ > ε

]⊆[ 4∑i=1

∣∣Xk∣∣ > ε

4

]⊆[ 4⋃i=1

[∣∣Xk∣∣ > ε

4

]]⊆

4⋃i=1

[∣∣Xk∣∣ > ε

4

]


we obtain Convergence (IV.26) if 6 for all but finitely many m ∈ N holds[∣∣Lm,(p)∞ − L∞,(p)∣∣] ⊆[∣∣∣∣Lm,(p)∞ −

∫(0,∞)

fm(s, Y m,(p)s , Zε,(p)s , Uε,(p)(s, ·)

)dCms

∣∣∣∣ > ε

4

]⋃[∣∣∣∣∫

(0,∞)

(fm − f∞)(s, Y m,(p)s , Zε,(p)s , Uε,(p)(s, ·)

)dCms

∣∣∣∣ > ε

4

](IV.28)

⋃[∣∣∣∣∫(0,∞)

f∞(s, Y m,(p)s , Zε,(p)s , Uε,(p)(s, ·)

)dCms −

∫(0,∞)

f∞(s, Y∞,(p)s , Zε,(p)s , Uε,(p)(s, ·)

)dC∞s

∣∣∣∣ > ε

4

]⋃[∣∣∣∣∫

(0,∞)

f∞(s, Y∞,(p)s , Zε,(p)s , Uε,(p)(s, ·)

)dC∞s − L∞,(p)∞

∣∣∣∣ > ε

4

]= ∅.

Therefore, we obtain the Convergence of the the sequence in (LS(p)∞ ) in probability, if we prove that the

first three sets of the right-hand side of (IV.28) are empty for all but finitely many m ∈ N. The fourthone is the empty set if (IV.27) is true.

The analogous decomposition can be done for (IV.25), where the | · | has to be substituted by themetric δJ1(R). Returning now to the uniform integrability that the sequences should satisfy, we will need

to prove that the family(

supt∈R+|Lk,(p)t |2)k∈N is uniformly integrable, which is a sufficient condition for

concluding both the Convergence (LS(p)) and (LS(p)∞ ).

From now on we fix an arbitrary subsequence(Lkl,(p)

)l∈N, resp.

(Lkl,(p)∞

)l∈N.

Let us end this part by collecting all the information we have available for the next subsections. Wewill state them as a remark so that it is easily referred. Moreover, for notational convenience we canassume that the sequence for which the forthcoming convergence are obtained P−almost surely is indexedby (klm)m∈N. This can be done without loss of generality, since we pass to a further subsequence finitelymany times.

Remark IV.23. (i) By Lemma IV.14 we have that there exist a δJ1(R)−convergent subsequence(Cklm

)m∈N as well as a δ|·|−convergent subsequence

(Cklm∞)m∈N.

(ii) By Condition (B7).(iii) we obtain in particular that

fklm(·, 0, 0,0

) J1(R)−−−−→ f∞(·, 0, 0,0

)P− a.s..

(iii) The Convergence (LS(p−1)), (LS(p−1)∞ ), which are assumed true as the induction assumption, allows

us to obtain that

Lklm ,(p−1) J1(R)−−−−→m→∞

L∞,(p−1) as well as Lklm ,(p−1)∞

|·|−−−−→m→∞

L∞,(p−1)∞ P− a.s..

(iv) In view of the discussion made in the main body of the proof of Theorem IV.10, see in particular

on p. 111, the validity of the Convergence (LS(p−1)) and (LS(p−1)∞ ) allows us to apply Theorem III.50 for

the martingale sequence (Mk,(p−1))k∈N. Therefore, we can apply Lemma I.106 because of the uniform

integrability of the sequence (∣∣Mk,(p−1)∞

∣∣2)k∈N in order to obtain that(

supt∈R+

∣∣Mk,(p−1)t

∣∣2)k∈N

is uniformly integrable. (IV.29)

Moreover, we obtain that the following convergence are true

(Mk,(p−1), 〈Zk,(p) ·Xk,〉, 〈Uk,(p) ? µk〉)(J1(R3),L2)−−−−−−−−→

k→∞(M∞,(p−1), 〈Z∞,(p) ·X∞,c〉, 〈U∞,(p) ? µ∞〉).

(IV.30)

and

(〈Zk,(p) ·Xk,, Xk,〉, 〈Uk,(p) ? µk, Xk,\〉)(J1(R2),L2)−−−−−−−−→

k→∞(〈Z∞,(p) ·X∞,c, X∞,c〉, 〈U∞,(p) ? µ∞, X∞,d〉).

(IV.31)

6For notational convenience we index the klm−element in the next expression simply by m.


(v) In view of (iv) we can assume that

(Mklm ,(p−1), 〈Zklm ,(p) ·Xklm ,〉, 〈Uklm ,(p) ? µklm 〉) J1(R3)−−−−−→m→∞

(M∞,(p−1), 〈Z∞,(p) ·X∞,c〉, 〈U∞,(p) ? µ∞〉) P− a.s.(IV.32)

and

(〈Zklm (p) ·Xklm, Xklm〉, 〈Uklm (p) ? µk, Xklm \〉) J1(R2)−−−−−→m→∞

(〈Z∞,(p) ·X∞,c, X∞,c〉, 〈U∞,(p) ? µ∞, X∞,d〉) P− a.s..(IV.33)

(vi) Recall (IV.15) and (IV.17), i.e. Y k,(p) = Mk,(p−1) − Lk,(p−1) for every k ∈ N. we claim that(supt∈R+

∣∣Y k,(p)t

∣∣2)k∈N

is uniformly integrable. (IV.34)

This can be concluded as follows. We have

supt∈R+

∣∣Y k,(p)t

∣∣2 ≤ 2 supt∈R+

∣∣Mk,(p−1)t

∣∣2 + 2 supt∈R+

∣∣Lk,(p−1)t

∣∣2 for every k ∈ N.

For the sequence(

supt∈R+

∣∣Lk,(p−1)t

∣∣2)k∈N

we can conclude its uniform integrability by arguing analo-

gously to Proposition IV.21 since (Γk,(p−1))k∈N has been assumed uniformly integrable; see (Step (p− 1) UI).Now, we can conclude our initial statement by means of Corollary I.29.(iv) and because of (IV.29).(vii) In view of Condition (B9).(iii), which implies that L∞,(p−1) is a continuous process, (iii), (v),

Proposition I.121 and Corollary I.122, we can conclude that

Y klm ,(p)J1(Rn)−−−−−→m→∞

Y∞,(p) P− a.s..

Remark IV.24. The purpose of the above remarks was not only to collect all the available information,but also to provide us with an Ωsub ⊂ Ω with P(Ωsub) = 1 such that the above properties hold for everyω ∈ Ωsub. In the next subsections whenever we say a property holds P−almost surely for a subsequenceindexed by (klm)m∈N, the reader understands for every ω ∈ Ωsub.

Now, we are ready to proceed to prove our claim under this convenient framework.

IV.5.1. Lusin approximation. In this section we provide a pathwise approximation for the fourthsummand of (IV.28). The reader may revise Notation II.2, Notation II.6 and Notation IV.5. We startwith a lemma which justifies the validity of the approximation we will obtain.

Lemma IV.25. Let Y , Z and U with the following properties

• Y : Ω× R+ → R is a process with cadlag and bounded ω−paths for ω ∈ Ωsub.• Z : Ωsub × R+ → R is such that Z(ω, ·) ∈ Cc(R+;R) for every ω.• U : Ωsub × R+ × R→ R is such that U(ω, ·) ∈ Cc,0(R+ × R;R) for every ω.

Then,

(i) For every ω ∈ Ωsub∫(0,∞)

∣∣Z(ω, s)∣∣2 d〈Xklm ,〉s(ω) <∞ for every m ∈ N and

∫(0,∞)

∣∣Z(ω, s)∣∣2 d〈X∞,c〉s(ω) <∞.

(ii) For every s ∈ R+, for every (ω, t) ∈ Ωsub ×R+ and for every m ∈ N the quantity∣∣∣∣∣∣Us(·)∣∣∣∣∣∣t,klm (ω)

is well-defined.(iii) For every (ω, s), (ω, t) ∈ Ωsub × R+ and for every m ∈ N

fklm(ω, t, Yt(ω), Z(ω, t), U(ω, t, ·)

)is well-defined and real number

as well as

fklm(ω, t, Yt(ω), Z(ω, s), U(ω, s, ·)

)is well-defined and real number.

Moreover, ∫(0,∞)

∣∣fklm (ω, t, Yt(ω), Z(ω, t), U(ω, t, ·))∣∣ dCkt (ω) <∞


as well as ∫(0,∞)

∣∣fklm(ω, t, Yt(ω), Z(ω, s), U(ω, s, ·))∣∣ dCk

t (ω) < ∞.

Proof. For each ω ∈ Ωsub Z(ω.·) and U(ω, ·) are Borel measurable as continuous functions. There-fore, no measurability issues arise.

(i) It is clear.

(ii) For the well-posedness of∣∣∣∣∣∣Us(·)

∣∣∣∣∣∣t,klm

(ω) it is sufficient to prove (recall its definition by (II.9))

that

KĎX

klm

t

(|Us(·)|2)< ∞ and

∣∣KĎXklm

t

(Us(·)

)∣∣2 < ∞.

By Assumption (C), for every m ∈ N the transition kernel KĎX

klm is such that it satisfies the properties(I.71), i.e. for every m ∈ N

KĎX

klm

t (ω; 0) = 0,

∫R

(|x|2 ∧ 1)KĎX

klm

t (ω; dx) ≤ 1 and ΔCĎX

klm (ω)KĎX

klm

t (ω;R) ≤ 1.

In the following, we will use the second property in conjunction with the fact that supp(U(ω)) is compactwith

inf‖z − w‖2, z ∈ supp(U(ω)) and w ∈ E0

> 0.

Let us define

Π :=x ∈ R, ∃s ∈ R+ such that (s, x) ∈ supp

(U(ω, ·)).

Now, we have

0 < inf(|t− s|2 + |x− y|2) 1

2 , (t, x) ∈ supp(U(ω, ·)) and (s, y) ∈ E0

≤ inf

(|t− s|2 + |x|2) 12 , (t, x) ∈ supp

(U(ω, ·)) and (s, y) ∈ R+ × 0

= inf|x|, ∃t ∈ R+ such that (t, x) ∈ supp(U(ω, ·))

= inf|x|, x ∈ ΠThe last property, in turn, implies that the function R x −→ (|x|2 ∧ 1)−11Π(x) ∈ R+ is bounded frombelow by 1 and from above by the real number

sup(|x|2 ∧ 1)−1, x ∈ Π

= inf

|x|2 ∧ 1, x ∈ Π−1

=(inf|x|, x ∈ Π ∧ 1

)−1< ∞. (IV.35)

On the other hand, we have in particular,∫R

[(|x|2 ∧ 1)1Π(x)

]K

ĎX∞s (ω; dx) ≤ 1, for every s ∈ R+. (IV.36)

Combining (IV.35) and (IV.36) we can conclude that

KĎX

klm

t

(Us(·)

)is a real-number. We can argue analogously for K

ĎXklm

t

(∣∣Us(·)∣∣2).

(iii) The Lipschitz property of fklm for every m ∈ N allows us to conclude.

Proposition IV.26. For every ε > 0, there exist Zε,(p) : Ωsub×R+ → R and Uε,(p) : Ωsub×R+×R → R

(which are defined ω-by-ω) with the following properties:

(i) Zε,(p)(ω, ·) ∈ Cc(R+;R) for every ω.

(ii) Uε,(p)(ω, ·)(ω, ·) ∈ Cc,0(R+ × R;R) for every ω.

(iii) The pair(Zε,(p), Uε,(p)

)7 satisfies∣∣∣∣ ∫

(0,∞)

[f∞(s, Y ∞,(p)

s , Zε,(p)s , Uε,(p)(s, ·))− f∞(s, Y ∞,(p)

s , Z∞,(p)s , U∞,(p)(s, ·))] dC∞

s

∣∣∣∣ < ε

4P− a.s..

7We will omit the ω in order to simplify notation. Moreover, we will denote Zε,(p)(ω, s) by Zε,(p)s when we do not

write the ω.


Proof. Let us fix an ε > 0. We deal initially with the Zε,(p)−approximation. We will simplify also the notation for X∞,c by

simply referring to it as Xc, since no confusion can arise at this point. We claim that we can determine

a Z such that Z(ω, ·) is continuous and of compact support which additionally satisfies P− a.s.∫(0,∞)

eβA∞s

∣∣Zs − Z∣∣2 d〈Xc〉s <ε2β

32. (IV.37)

We remind the reader that the process 〈Xc〉 is increasing, continuous (recall (B1)), G∞−predictablewhose limit limt→∞〈Xc〉∞ is a real-number P−almost surely, due to the fact that Xc ∈ H2(G∞;R). Inparticular, 〈Xc〉(ω) is B(R+)−measurable for every ω ∈ Ωsub. Therefore, we can associate to 〈Xc〉(ω) afinite measure on

(R+,B(R+)

)for every ω ∈ Ωsub. We denote the associated measure by 〈Xc〉 in order

to keep notation simple. Since (R+, δ|·|) is Polish we have that the measure 〈Xc〉 is a Radon measure; seeBogachev [10, 7.1.1. Definition, 7.1.7. Theorem]. Moreover, (R+, δ|·|) is locally compact, which allows usto apply Lusin’s Theorem, see [10, 7.1.13. Theorem]. Therefore, by using standard arguments we obtainfinally that the space Cc(R+) is dense8 on

L2β,A∞

(〈Xc〉(ω)

):=f :(R+,B(R+)

)→(R,B(R)

),

∫(0,∞)

eβA∞s |fs|2d〈Xc〉(ω)

for every ω ∈ Ωsub.

We deal now with the Uε,(p)−approximation. The arguments are more or less analogous to theprevious case. We ease notation by assigning also the symbol U to the G∞−predictable random functionU∞,(p). We start with the fact that ν∞ as well as q ∗ ν∞∞ are well-defined for every ω ∈ Ωsub. In view

of Remark IV.7, we will assume that ν∞(ω) is a measure on(E,B(E)

). The space E is metric, but

not Polish, since it is not complete. Therefore, we have that the measure ν∞(ω; d, s,dx) is regular; seeBogachev [10, 7.1.1 Definition, 7.1.7. Theorem]. However, we can use that it is locally compact and thatν∞ is tight, as a σ−compact measure. Now, Remark IV.7 allows us to use indifferently the measurable

spaces(E,B(E)

)and

(R+ × R,B(R+) ⊗ B(R)

), since the measure ν∞(ω) is for every ω ∈ Ωsub well-

defined on both with zero mass on E0. After this discussion, we can apply Lusin’s Theorem, see [10,7.1.13. Theorem], in order to obtain (using standard arguments as described before) that the spaceCc,0(R+ × R;R) is dense in

L2βA∞

(ν∞(ω)

):=f :(R+ × R,B(R+)⊗ B(R)

)→(R,B(R)

), (eβA

∞|f |2) ∗ ν∞∞(ω)

for every ω ∈ Ωsub.

Therefore, we can find for every ω ∈ Ωsub a function U(ω, ·) ∈ Cc,0(R+ × R;R) such that∫(0,∞)

eβA∞s (ω)

(∣∣∣∣∣∣Us(·)− Us(·)∣∣∣∣∣∣s,∞(ω))2

dC∞s (ω) <ε2β

32. (IV.38)

Now we are ready to provide the required estimation in view of the following set inclusions[∣∣∣∣ ∫(0,∞)

f∞(s, Y∞,(p)s , Zε,(p)s , Uε,(p)(s, ·)

)dC∞s − L∞,(p)∞

∣∣∣∣ ≥ ε

4

]=

[∣∣∣∣ ∫(0,∞)

f∞(s, Y∞,(p)s , Zε,(p)s , Uε,(p)(s, ·)

)dC∞s − L∞,(p)∞

∣∣∣∣2 ≥ ε2

16

]⊆

[∫(0,∞)

∣∣∣f∞(s, Y∞,(p)s , Zε,(p)s , Uε,(p)(s, ·))− f∞

(s, Y∞,(p)s , Z∞,(p)s , U∞,(p)(s, ·)

)∣∣∣2 dC∞s ≥ε2

16

](II.16)

⊆(II.25)

[1

β

∫(0,∞)

eβA∞s

[∣∣c∞s (Zε,(p)s − Zs)∣∣2 +

∣∣∣∣∣∣Uε,(p)s (·)− Us(·)∣∣∣∣∣∣s,∞

]dC∞s ≥

ε2

16

]9

⊆[

1

β

∫(0,∞)

eβA∞s

∣∣c∞s (Zε,(p)s − Zs)∣∣2 dC∞s ≥

ε2

32

]+

[1

β

∫(0,∞)

eβA∞s

∣∣∣∣∣∣Uε,(p)s (·)− Us(·)∣∣∣∣∣∣2s,∞ dC∞s ≥

ε2

32

](IV.37)

⊆(IV.38)

∅.

8We prove initially that Cc(R+) is dense in the space of simple functions, which is dense in the space L2(〈Xc〉(ω)

).


IV.5.2. Weak convergence of the integrators. In this section we are going to use partially theinformation provided in Remark IV.23 in order to prove that P−almost surely holds∫

(0,·]f∞(s, Y

klm ,(p)s , Zε,(p)s , Uε,(p)(s, ·)

)dC

klms

J1−−−−→m→∞

∫(0,·]

f∞(s, Y∞,(p)s , Zε,(p)s , Uε,(p)(s, ·)

)dC∞s

as well as∫(0,∞)

f∞(s, Y

klm ,(p)s , Zε,(p)s , Uε,(p)(s, ·)

)dC

klms

|·|−−−−→m→∞

∫(0,∞)

f∞(s, Y∞,(p)s , Zε,(p)s , Uε,(p)(s, ·)

)dC∞s .

In other words, we will prove that the third set of the union of (IV.28) is finally empty. These resultsare presented in Proposition IV.32, which is actually an application of Theorem A.14. More precisely, wewill apply Proposition A.16 and Corollary A.18. The aforementioned theorem provides a characterisationof the weak convergence of finite measures on the positive real-line based on relatively compact subsetsof D. This characterisation, which is new to the best of author’s knowledge, generalises the classicalcharacterisation based on relatively compact sets of the space of continuous functions. One could commentthat this is not surprising, since the relatively compact sets of D under the δJ1

−metric admit an Arzela–Ascoli-type characterisation; see Theorem I.111.

Therefore, in order to proceed we have to provide the preparatory results for applying Theorem A.14.Recall that in Remark IV.24 we have fixed an Ωsub with P(Ωsub) = 1.

Remark IV.27. For notational convenience, we will omit the superscripts ε and (p) from Zε,(p), Uε,(p).

Hence, in the rest of the Section IV.5, we assume that a fixed 0 < ε < 1β

is given and Z, U satisfy the

properties of Proposition IV.26. Moreover, the klm−element of a sequence indexed on k ∈ N will besimply denoted as indexed by m.

Lemma IV.28. For every m ∈ N, we define

f∞t [Y m,(p)](ω) := f∞(ω, t, Y

m,(p)t (ω), Z(ω, t), U(ω, t, ·)

)for ω ∈ Ωsub and for t ∈ R+. Then, for every t ∈ R+

lims↓t

f∞(ω, s, Y m,(p)s (ω), Z(ω, s), U(ω, s, ·)

)= f∞t [Y m,(p)](ω)

and

f∞t−[Y m,(p)](ω) = f∞(ω, t−, Y m,(p)t− (ω), Z(ω, t−), U(ω, t−, ·)

)= f∞

(ω, t−, Y m,(p)t− (ω), Z(ω, t), U(ω, t, ·)

).

In other words, f∞· [Y m,(p)](ω) ∈ D(R) for every ω ∈ Ωsub and every m ∈ N..

Proof. Let us fix an m ∈ N and (ω, t) ∈ Ωsub×R+. Firstly, the reader may recall Lemma IV.25 forthe well-posedness of f∞t [Y m,(p)](ω). We distinguish now two cases.

Let us assume that s ↓ t. We will prove that f∞s [Y m,(p)]|·|−−→s↓t

f∞t [Y m,(p)]. To this end, we calculate

an upper bound for |f∞s [Y m,(p)]− f∞t [Y m,(p)]|2.

|f∞s [Y m,(p)]− f∞t [Y m,(p)]|2

≤ 2|f∞s [Y m,(p)]− f∞(s, Y

m,(p)t , Zt, U(t, ·)

)|2 + 2|f∞

(s, Y

m,(p)t , Zt, U(t, ·)

)− f∞t [Y m,(p)]|2

(II.16)

≤ 2r∞s |Y m,(p)s − Y m,(p)t |2 + 2θ∞,cs |c∞s (Zs − Zt)|2 + 2θ∞,ds

∣∣∣∣∣∣Us(·)− Ut(·)∣∣∣∣∣∣2s,∞+ 2|f∞

(s, Y

m,(p)t , Zt, U(t, ·)

)− f∞t [Y m,(p)]|2.

(IV.39)

First of all recall that by Condition (B7).(iv) the bounds r∞, θ∞,c and θ∞,d are P−almost surelybounded, while the process

∣∣d〈X∞,c〉s/dC∞s |2 has locally bounded paths P−almost surely. Let us as-

sociate to the P−almost surely bounded processes the stochastic bounds C(r∞), C(θ∞,c) and C(θ∞,d).

9We have used the fact that θ∞,c, θ∞,d ≤ α∞; see (II.49).


Since the process d〈X∞,c〉s/dC∞

s has locally bounded paths, for the point (ω, t) there exists a δ = δ(ω, t)such that

C(RN;B(t, δ)

)10 := sup

s∈(t−δ,t+δ)

∣∣∣d〈X∞,c〉sdC∞

s

∣∣∣2 < ∞. (IV.40)

The first summand convergences to 0 because Y m,(p) is cadlag P−almost surely and 0 ≤ r∞s ≤C(r∞) < ∞ P−almost surely.

The second summand can be written for s such that |s− t| < δ as

2θ∞,cs |c∞s (Zs − Zt)|2 ≤ 2C(θ∞,c)C

(RN;B(t, δ)

)∣∣Zs − Zt

∣∣2 −−−→s↓t

0, (IV.41)

since Z is (uniformly) continuous.For the third summand we collect initially some facts which will help us to conclude. Recall that the

pair (ĎX∞, C∞) satisfies Assumption (C); see on p. 49. Therefore, we have in particular that∫R

(|x|2 ∧ 1)KĎX∞s (ω; dx) ≤ 1, for every s ∈ R+,

which in turn implies for

Π :=x ∈ R, ∃s ∈ R+ such that (s, x) ∈ supp(U)

that ∫

R

[(|x|2 ∧ 1)1Π(x)

]K

ĎX∞s (ω; dx) ≤ 1, for every s ∈ R+. (IV.42)

On the other hand, supp(U) is a compact subset of the open E = (R+ ×R)\ E0; recall Proposition IV.26and Notation IV.5. This implies that

inf|z − w|, z ∈ supp(U) and w ∈ E0

> 0, (IV.43)

as the intersection of the compact set supp(U) with the closed set E0 is empty. Using (IV.43) and treating

for the first two forthcoming lines the sets supp(U) and E0 as subsets of the product R+ × R we have

0 < inf(|t− s|2 + |x− y|2) 1

2 , (t, x) ∈ supp(U) and (s, y) ∈ E0

≤ inf

(|t− s|2 + |x|2) 12 , (t, x) ∈ supp(U) and (s, y) ∈ R+ × 0

= inf|x|, ∃t ∈ R+ such that (t, x) ∈ supp(U)= inf|x|, x ∈ Π

The last property, in turn, implies that the function R x −→ (|x|2 ∧ 1)−11Π(x) ∈ R+ is bounded frombelow by 1 and from above by the real number

sup(|x|2 ∧ 1)−1, x ∈ Π

= inf

|x|2 ∧ 1, x ∈ Π−1

=(inf|x|, x ∈ Π ∧ 1

)−1< ∞. (IV.44)

Using these observations and recalling Notation II.2 as well as that C∞ is continuous by (B9).(iii), wehave for the third summand

0 ≤ 2θ∞,ds

∣∣∣∣∣∣Us(·)− Ut(·)∣∣∣∣∣∣2

s,∞ ≤ 2C(θ∞,d) KĎX∞s

(|Us(·)− Ut(·)|2)

= 2C(θ∞,d)

∫R

(|U(s, x)− U(t, x)|2)KĎX∞s (dx)11

(IV.42)

≤ 2C(θ∞,d) supx∈Π

[(|x|2 ∧ 1)−1 |U(s, x)− U(t, x)|2

]≤ 2C(θ∞,d) sup

x∈Π(|x|2 ∧ 1)−1 sup

x∈Π|U(s, x)− U(t, x)|2 −−−→

s↓t0. (IV.45)

For the conclusion, we have used the fact that U is uniformly continuous (as continuous with compactmetrisable support) and that |(t, x)− (s, x)| = |t− s| for every s, t ∈ R+ and x ∈ R.

Finally, for the fourth summand we have only to make use of the right-continuity of the path(f∞(ω, s, Y m,(p)

t (ω), Z(ω, t), U(ω, t, ·)))s∈R+

;

10We abstain from writing the dependence on ω, which is clear in view of the preceding comments.11We have again abused notation by omitting the ω.


recall Condition (B7).(i) and the fact that U(t, ·) ∈ Cc(R).

Let us assume now that s ↑ t. Then, completely analogously to the previous case we obtain

|f∞s [Y m,(p)]− f∞(t−, Y m,(p)t− , Zt, U(t, ·)

)|2

(II.16)

≤ 2r∞s |Y m,(p)s − Y m,(p)t− |2 + 2θ∞,cs |c∞s (Zs − Zt)|2 + 2θ∞,ds

∣∣∣∣∣∣Us(·)− Ut(·)∣∣∣∣∣∣2s,∞+ 2|f∞

(s, Y

m,(p)t− , Zt, U(t, ·)

)− f∞

(t−, Y m,(p)t− , Zt, U(t, ·)

)|2.

For the first three summands we can argue as in the previous case. For the fourth summand, we use theleft-continuity of the path (

f∞(ω, s, Y

m,(p)t− (ω), Z(ω, t), U(ω, t, ·)

))s∈R+

,

which enables us to obtain the equality

f∞(ω, t−, Y m,(p)t− (ω), Z(ω, t), U(ω, t, ·)

)= lim

s↑tf∞(ω, s, Y

m,(p)t− (ω), Z(ω, t), U(ω, t, ·)

).

In view of the last comment, we can conclude.

Remark IV.29. Using the same arguments we can prove also for everym ∈ N that(fmt [Y m,(p)](ω)

)t∈R+

∈D(R), where for every t ∈ R+

fmt [Y m,(p)](ω) := fm(ω, t, Y

m,(p)t (ω), Z(ω, t), U(ω, t, ·)

). (IV.46)

We inform the reader that we will adopt the notation of Lemma IV.28 for the rest of the subsection.

Lemma IV.30. It holds f∞[Y m,(p)](ω)J1(R)−−−−→m→∞

f∞[Y∞,(p)](ω) for every ω ∈ Ωsub.

Proof. By Remark IV.23.(vii) we have that Y m,(p)J1−−−−→

m→∞Y∞,(p) P−almost surely. Therefore, by

Theorem I.109 there exists for almost every ω a sequence (λmω )m∈N ⊂ Λ (see Definition I.108) such that

supt∈R+

|λmω (t)− t| −−−−→m→∞

0 and supt∈[0,N ]

∣∣Y m,(p)λmω (t) (ω)− Y∞,(p)t (ω)∣∣ −−−−→m→∞

0 for every N ∈ N. (IV.47)

Using again Theorem I.109, it is sufficient to prove that

supt∈[0,N ]

∣∣f∞λmω (t)[Ym,(p)](ω)− f∞t [Y∞,(p)](ω)

∣∣ −−−−→m→∞

0 for every N ∈ N. (IV.48)

To this end, using analogous to (IV.39) arguments we obtain (where we omit the dependence in ω to easenotation)∣∣f∞λm(t)[Y

m,(p)]− f∞t [Y∞,(p)]∣∣2

≤ 2∣∣f∞λm(t)[Y

m,(p)]− f∞(λm(t), Y

∞,(p)t , Zt, U(t, ·)

)∣∣2 + 2∣∣f∞(λm(t), Y

∞,(p)t , Zt, U(t, ·)− f∞t [Y∞,(p)]

∣∣2(II.16)

≤ 2r∞λm(t)

∣∣Y m,(p)λm(t) − Y∞,(p)t

∣∣2 + 2θ∞,cλm(t)

∣∣c∞λm(t)(Zλm(t) − Zt)∣∣2

+ 2θ∞,dλm(t)

∣∣∣∣∣∣Uλm(t)(·)− Ut(·)∣∣∣∣∣∣2λm(t),∞ + 2

∣∣f∞(λm(t), Y∞,(p)t , Zt, U(t, ·)

)− f∞t [Y m,(p)]

∣∣2.Let us, now, fix an N ∈ N. Using the above bound, we obtain (IV.48) if we can prove that the

following hold

• supt∈[0,N ]

∣∣Y m,(p)λmω (t) (ω)− Y∞,(p)t (ω)∣∣2 −−−−→

m→∞0, since r∞· (ω) is bounded. But this is true by Conver-

gence (IV.47).

• supt∈[0,N ]

∣∣c∞λm(t)(Zλm(t) − Zt)∣∣2 −−−−→

m→∞0, in view of the P−almost surely boundedness of θ∞,c· .

Name C(θ∞,c) the bound of θ∞,c· . Arguing as in (IV.40) and using that the set [0, N + 1] is compact, wecan determine12 a finite bound C(RN; [0, N + 1]) of d〈X∞,c〉s

/dC∞s on [0, N + 1], i.e.

C(RN; [0, N + 1]) := supt∈[0,N+1]

d〈X∞,c〉sdC∞s

.

12Since d〈X∞,c〉s/

dC∞s is locally bounded, using the notation of (IV.40) we have that⋃t∈(0,N+1](t− δ(ω, t) ∧ 0, t+

δ(ω, t))∪[0, δ(0, ω)

)is an open cover of [0, N + 1] in R+. The compactness of [0, N + 1] implies that there is a finite cover.

Therefore, we can use the maximum of the bounds which associate to the finite cover.


Therefore, analogously to (IV.41) we obtain (we omit ω)

supt∈[0,N ]

(2θ∞,c

λm(t)|c∞λm(t)(Zs − Zt)|2)≤ 2C(θ∞,c)C

(RN; [0, N + 1]

)sup

t∈[0,N ]

∣∣Zλm(t) − Zt

∣∣2, (IV.49)

when λm(t) ∈ [0, N+1]. Now, U is uniformly continuous, as continuous with compact metrisable support.

In other words, for the given ε > 0 (recall Remark IV.27) there exists a δ > 0 such that∣∣Zu − Zw

∣∣ < εwhenever |u−w| < δ. Without loss of generality, we can assume that δ ≤ 1. On the other hand, in view of(IV.47), for every given δ > 0 there exists an M ∈ N such that supt∈[0,N ] |λm(t)− t| < δ for every m ≥ M.

Combining the last two properties, we can finally obtain for every t ∈ [0, N ] that λm(t) ∈ [0, N + 1] forall but finitely many m and, moreover,

supt∈[0,N ]

∣∣Zλm(t) − Zt

∣∣2 −−−−→m→∞ 0,

which implies the required one, in view of (IV.49).

•∣∣∣∣∣∣Uλm(t)(·)− Ut(·)

∣∣∣∣∣∣2λm(t),∞ −−−−→

m→∞ 0, in view of the P−almost surely boundedness of θ∞,d· . Name

C(θ∞,d) the bound of θ∞,d. Arguing as in (IV.45) we can obtain (we omit the ω)

0 ≤ supt∈[0,N ]

2θ∞,dλm(t)

∣∣∣∣∣∣Uλm(t)(·)− Ut(·)∣∣∣∣∣∣2

λm(t),∞ ≤ 2C(θ∞,d) supt∈[0,N ]

KĎX∞λm(t)

(|Uλm(t)(·)− Ut(·)|2)

= 2C(θ∞,d) supt∈[0,N ]

∫R

(|U(λm(t), x)− U(t, x)|2)KĎX∞λm(t)(dx)

(IV.42)

≤ 2C(θ∞,d) supt∈[0,N ]

supx∈Π

[(|x|2 ∧ 1)−1 |U(λm(t), x)− U(t, x)|2

](IV.44)

≤ 2C(θ∞,d) supx∈Π

(|x|2 ∧ 1)−1 supt∈[0,N ]

supx∈Π

|U(λm(t), x)− U(t, x)|2 −−−−→m→∞ 0. (IV.50)

We have used again the uniform continuity of U as well as the fact that limm→∞ supt∈R+|λm(t)− t| = 0.

• supt∈[0,N ]

∣∣f∞(λm(t), Y∞,(p)t , Zt, U(t, ·)) − f∞

t [Y m,(p)]∣∣2 −−−−→

m→∞ 0. But, this is immediate in view

of Condition (B7).(ii) and the boundedness of(∣∣Y ∞,(p)

t

∣∣)t∈R+

and (∣∣Zs

∣∣)s∈R+ . For the former set recall

Remark IV.23.(vi), while for the latter set it is immediate by the continuity of Z and the compactness of

its support. Therefore, there exist compact sets K1 ⊂ R and K2 ⊂ R such that (Y∞,(p)t )t∈R+

⊂ K1 and

(Zt)t∈R+⊂ K2 and we can indeed call (B7).(ii).

Lemma IV.31. For compact K1,K2 ⊂ R, and U ∈ Cc,0(R+ × R;R) it holds

supt∈R+

∣∣f∞(t, y, z, U(t, ·))∣∣, (y, z) ∈ K1 ×K2

< ∞ P− a.s..

Proof. Let K1,K2 ⊂ R be compact and U ∈ Cc,0(R+ × R;R). The claim is immediate by theLipschitz property of f∞, Condition (B7).(iv), the boundedness of K1,K2 as well as of the boundednessof the image of U . Proposition IV.32. It holds∫

(0,·]f∞(s, Y klm ,(p)

s , Zε,(p)s , Uε,(p)(s, ·)) dCklm

sJ1−−−−→

m→∞

∫(0,·]

f∞(s, Y ∞,(p)s , Zε,(p)

s , Uε,(p)(s, ·)) dC∞s

as well as∫(0,∞)

f∞(s, Y klm ,(p)s , Zε,(p)

s , Uε,(p)(s, ·)) dCklms

|·|−−−−→m→∞

∫(0,∞)

f∞(s, Y ∞,(p)s , Zε,(p)

s , Uε,(p)(s, ·)) dC∞s .

Proof. We will apply Proposition A.16 and Corollary A.18. In view of Remark IV.23.(i), Lemma A.13and Proposition A.12, we have that the associated to (Cm)m∈N

measures converge weakly. In the previouslemma we have proven that we have the the sequence of the integrands is δJ1(R)−convergent. We have

only to prove that the familyf∞t [Y m,(p)](ω), t ∈ R+ and m ∈ N

is bounded for every ω ∈ Ωsub. In

view of Lemma IV.31 we need only to prove thatY

m,(p)t (ω), t ∈ R+ and m ∈ N

is bounded for every

ω ∈ Ωsub.


To this end we are going to apply Lemma A.19. Let us verify its requirements. The first one issatisfied by Remark IV.23.(vi), while (ii) is satisfied because (Y m,(p))m∈N is a compact set of D(R) asJ1(R)−convergent sequence Remark IV.23.(vii); see also Theorem I.111. For the third requirement, wehave by Remark IV.23.(vi) that

E[

lim sup(m,t)→(∞,∞)

∣∣Y m,(p)t

∣∣] = E[

limn→∞

∣∣Y mn,(p)tn

∣∣]13 ≤ E[

limn→∞

supt∈R+

∣∣Y mn,(p)t

∣∣]≤ lim inf

n→∞E[

supt∈R+

∣∣Y mn,(p)t

∣∣] ≤ supk∈N

E[

supt∈R+

∣∣Y k,(p)t

∣∣] <∞.Therefore, P

([lim sup(m,t)→(∞,∞)

∣∣Y m,(p)t

∣∣ =∞])

= 0 which allows us to conclude that

P([

supm∈N|Y m,(p)|∞ <∞

])= 1. (IV.51)

Using that P([|Y∞,(p)|∞ <∞

])= 1, we conclude also that supm∈N

∣∣Y m,(p)(ω)∣∣∞ <∞ for every ω ∈ Ωsub,

without loss of generality.

IV.5.3. The drivers converge uniformly on compacts. In this subsection, which actually con-sists of a single result, we will justify that for all but finally many m ∈ N holds∣∣∣∣∫

(0,∞)

(fm − f∞)(s, Y m,(p)s , Zε,(p)s , Uε,(p)(s, ·)

)dCms

∣∣∣∣ < ε

4. (IV.52)

Lemma IV.33. The Inequality (IV.52) is true for all but finitely many m ∈ N.

Proof. In view of Condition (B7).(iii) and the compactness of the support of Z(ω, ·), it is sufficient

to prove that the set∣∣Y m,(p)t |, t ∈ R+ and m ∈ N

is bounded. But, this is proven in (IV.51). Then,

since supm∈N(Cm∞) <∞ we can conclude.

IV.5.4. The predictable quadratic variations converge. In this subsection we are going toprove that for all but finitely many m ∈ N holds∣∣∣∣Lm,(p)∞ −

∫(0,∞)


)dCms

∣∣∣∣ < ε

4,

in other words we are going to prove that the second set of the right hand side of (IV.28) is finallyempty. At this part the Condition (B7).(iii) in conjunction with Convergence (IV.32) will provide usthe required result. In the whole subsection the measures are treated as Borel measures.

We start with a lemma for the convergence of the sequence (Am)m∈N.

Lemma IV.34. The sequence (Am)m∈N is ‖ · ‖∞−convergent. Consequently,

eβAm ‖·‖∞−−−−→ eβA

∞.

Proof. Recall that by definition Am· :=∫

(0,·](αms )2 dCms ∈ V+(Gm;R) for every m ∈ N. Therefore,

we obtain the locally uniform convergence once we prove the pointwise convergence for t ∈ R+; see Jacodand Shiryaev [41, Proposition VI.2.15 c)]. We can then conclude the uniform convergence by Lemma A.13.In total, we need to prove that Amt −→ A∞t for every t ∈ [0,∞].

By Condition (B7).(iv) and by the definition of (αm· )2 := max√

rm· , θc· , θ

d·

(see (F4) on p. 58)

for every k ∈ N we have αm·‖·‖∞−−−−→ α∞· . Now, recall that α∞ is continuous C∞−almost everywhere (see

(B7).(iii)) and the weak convergence of the measures associated to the sequence (Cm)m∈N in order toapply the characterisation of Mazzone [51, Theorem 1]. In view of the above, we obtain for t ∈ R+∣∣Amt −A∞t ∣∣ =

∣∣∣∣ ∫(0,·]

(αms ) dCms −∫

(0,·](α∞s ) dCms

∣∣∣∣+

∣∣∣∣ ∫(0,·]

(α∞s ) dCms −∫

(0,·](α∞s ) dC∞s

∣∣∣∣≤∫

(0,·]

∣∣∣(αms )− (α∞s )∣∣∣ dCms +

∣∣∣∣ ∫(0,·]


(0,·](α∞s ) dC∞s

∣∣∣∣≤∥∥αm − α∞∥∥∞ sup

m∈NCm∞ +

∣∣∣∣ ∫(0,·]


(0,·](α∞s ) dC∞s

∣∣∣∣ −−−−→m→∞0.

13The equality holds for some subsequence((mn, tn)

)n∈N, which may depend on the path, such that (mn, tn) →

(∞,∞).


Analogously we can prove the convergence for t =∞.

The previous lemma in conjunction with Convergence (IV.32) - (IV.33) provides the following results.

Lemma IV.35. The following convergence are true∫(0,·]

eβAms (Zm,(p)s )2d〈Xm,〉s

J1(R)−−−−→m→∞

∫(0,·]

eβA∞s (Z∞,(p)s )2d〈X∞,c〉s (IV.53)

and ∫(0,·]

eβAms Zm,(p)s d〈Xm,〉s

J1(R)−−−−→m→∞

∫(0,·]

eβA∞s Z∞,(p)s d〈X∞,c〉s. (IV.54)

The same is true when the domain of integration is the whole positive real line and the metric is substitutedby | · |. Moreover, for every R : R+ → R bounded function, which can be approximated uniformly bypiecewise-constant functions, the following convergence is true∫

(0,·]eβA

ms RsZ

m,(p)s d〈Xm,〉s

J1(R)−−−−→m→∞

∫(0,·]

eβA∞s RsZ

m,(p)s d〈X∞,c〉s. (IV.55)

The same is true when the domain of integration is the whole positive real line and the metric is substitutedby | · |. In particular, R can be chosen to be continuous with compact support.

Proof. Observe that in each case, the limit-process is continuous. Therefore, the convergence underSkorokhod topology coincides with the locally uniform topology, and the same is true for (IV.32) -(IV.33), which are special case of (IV.53) and (IV.54) respectively. So, we will use the fact that theaforementioned convergence hold under the locally uniform topology. Now, we can prove the claim usingthe same argument as in the proof of Lemma IV.34. More precisely, in view of Convergence (IV.32)- (IV.33), we use the associativity of the Lebesgue–Stieltjes integral in order to prove initially that

the required convergence are true when their integrands are multiplied by eβA∞. Now, in view of the

boundedness of (Am)m∈N by (B6), we can conclude the uniform convergence of (eβAm

)m∈N to eβA∞.

Observe that the sequence (eβAm

)m∈N remains uniformly bounded. Then, using these last convergence,we use the fact that all measures are finite in order to conclude (IV.53) and (IV.54).

For the Convergence (IV.55), in view of (IV.33), we can repeat the previous arguments for the case

(eβAm

S)m∈N, where S is piecewise constant. Now, we can conclude again the general case, for the classdescribed in the statement since the convergence of the integrands will be under the uniform topology.

Lemma IV.36. The following convergence are true∫(0,·]×R

eβAms(Um,(p)(s, x)

)2νm(ds,dx)−

∑s≤·

eβAms

(∫RU(s, x)m,(p) νm(s × dx)

)2

J1(R)−−−−→m→∞

∫(0,·]×R

eβA∞s(U∞,(p)(s, x)

)2ν∞(ds,dx)

(IV.56)

and ∫(0,·]×R

eβAms Um,(p)(s, x)x νm(ds,dx)

−∑s≤·

eβAms

∫RUm,(p)(s, x) νm(s × dx)

∫Rx νm(s × dx)

J1(R)−−−−→m→∞

∫(0,·]×R

eβA∞s U∞,(p)(s, x)ν∞(ds,dx)

(IV.57)

The same is true when the domain of integration is the whole positive real line and the metric is substituted

by | · |. Moreover, for every S : E → R bounded function which vanishes outside of a compact set of E


and which is the uniform limit of piecewise constant functions,14 the following convergence is true∫(0,·]

eβAms Um,(p)(s, x)S(s, x)νm(ds,dx)

−∑s≤·

eβAms

∫RU(s, x)m,(p) νm(s × dx)

∫RS(s, x) νm(s × dx)

J1(R)−−−−→m→∞

∫(0,·]

eβA∞s U∞,(p)(s, x)S(s, x) ν∞(ds,dx)

(IV.58)

The same is true when the domain of integration is the whole positive real line and the metric is substitutedby | · |. In particular, the function S can be chosen to lie in Cc,0(R+ × R;R).

Proof. The convergence (IV.56) and (IV.57) are simply the translation of (IV.32) - (IV.33). The(IV.58) needs to be treated in a more careful way.

Lemma IV.37. The following inequality is true for all but finitely many m ∈ N∣∣∣∣Lm,(p)∞ −∫

(0,∞)

fm(s, Y m,(p)s , Zs, U(s, ·)

)dCms

∣∣∣∣ < ε

4.

Proof. Using the Cauchy–Schwartz Inequality as in (II.25) we obtain∣∣∣∣Lm,(p)∞ −∫

(0,∞)


)dCms

∣∣∣∣2=

∣∣∣∣∫(0,∞)

[fm(s, Y m,(p)s , Zm,(p)s , Um,(p)(s, ·)

)− fm

(s, Y m,(p)s , Zs, U(s, ·)

)]dCms

∣∣∣∣2=

∫(0,∞)

eβAms (αms )−2

∣∣fm(s, Y m,(p)s , Zm,(p)s , Um,(p)(s, ·))− fm

(s, Y m,(p)s , Zs, U(s, ·)

)∣∣2 dCms

(II.49)

≤(II.16)

∫(0,∞)

eβAms

[∣∣cms (Zm,(p)s − Zs)∣∣2 +

∣∣∣∣∣∣Um,(p)s (·)− Us(·)∣∣∣∣∣∣s,k

]dCms

=

∫(0,∞)

eβAms (Zm,(p)s − Zs)2 d〈Xm,〉s

+

∫(0,∞)

eβAms

(Um,(p)(s, x))2 − 2Um,(p)(s, x)U(s, x) + (U(s, x))2

dνm(ds,dx)

−∑s>0

eβAms

(∫RU(s, x)m,(p) νm(s × dx)

)2

+

∫RUm,(p)(s, x) νm(s × dx)

∫RU(s, x)m,(p) νm(s × dx)

+

(∫RUm,(p)(s, x) νm(s × dx)

)2By Lemma IV.35 and Lemma IV.36 we have that the right hand side of the above inequality convergesto the value∫

(0,∞)

eβA∞s (Z∞,(p)s − Zs)2 d〈X∞,c〉s +

∫(0,·]

eβA∞s(U∞,(p)(s, x)− U(s, x)

)2ν∞(ds,dx)

(IV.37)<

(IV.38)

εβ2

64.

Now recall that we have assumed (without loss of generality) in Remark IV.27 that ε < 1β

. Therefore,

we can conclude the claim of the lemma.

14In this case a piecewise constant function f if of the form f =∑pi=1 fi1Πi , where (Πi)i=1,...,p is a pairwise disjoint

family of rectangular subsets of E.


IV.5.5. Uniform integrability. For the following the reader may recall Notation IV.12. In thissubsection we will prove that the statement (Step (p) UI) is true i.e.

(Γk,(p))k∈N and

(∆k,(p))k∈N are

uniformly integrable. This in turn will ensure that the sequence(

supt∈R+

∣∣Lk,(p)t

∣∣2)k∈N is uniformly

integrable and they will allow to proceed with the induction. Recall that we need the uniform integrability

of(

supt∈R+

∣∣Lk,(p)t

∣∣2)k∈N in order to apply Vitali’s Theorem and finally prove that the Convergence

(LS(p)) and (LS(p)∞ ) hold in L2−mean and not only in probability.

For the following lemmata we will make use of the pathwise estimates obtained in Lemma II.18. Aswe did in Lemmata IV.19 and IV.20, in view of Condition (B6) we will substitute the coefficients of theaforementioned inequalities with a universal constant, say C. After these comments, we can proceed.


supt∈R+

∣∣Mk,(p)t −Mk,(p)

0

∣∣2)k∈N

is uniformly integrable. In particular,

the sequence(∆k,(p))k∈N is uniformly integrable.

Proof. The arguments are very similar to the one used to prove Lemma IV.19. However, we provideall the details for the sake of completeness.

The martingale Mk,(p), for k ∈ N, was defined in (IV.15) for k = ∞ and in (IV.17) for k ∈ N.However, we will use the equivalent forms (IV.16) and (IV.18). Now, we proceed to prove the first claimof the lemma. We will make use of the pathwise estimates in Lemma II.18 and in particular (II.39). Wehave, now, for every k ∈ N that

supt∈R+


0

∣∣2≤ C sup

t∈R+

E[∣∣ξk∣∣2 +

∫(0,∞)

eβAks

∣∣fk(s, Y k,(p−1)s , Z

k,(p−1)s , Uk,(p−1)(s, ·)

)∣∣2(αks )2

dCks

∣∣∣∣Gt]+ C

∫(0,∞)

eβAks

∣∣fk(s, Y k,(p−1)s , Z

k,(p−1)s , Uk,(p−1)(s, ·)

)∣∣2(αks )2

dCks

≤ C supt∈R+

E[∣∣ξk∣∣2 + Γk,(p−1)

∣∣∣∣Gt].+ CΓk,(p−1)

The right hand side is uniformly integrable; this is true in view of Lemma I.106 and the uniform integra-

bility of the sequence(∣∣ξk∣∣2)

k∈N and of(Γk,(p)

)k∈N.

In view of the uniform integrability of the sequence(

supt∈R+


0

∣∣2)k∈N

we can apply

Theorem I.55 in order to obtain the uniform integrability of(〈Mk,(p) −Mk,(p)

0 〉∞)k∈N. Then, we have

the required property for the sequence(∆k,(p))k∈N.

Lemma IV.39. The sequence(Γk,(p))k∈N is uniformly integrable.

Proof. We will make use of Lemma II.18 and in particular the Let k ∈ N. Then, we have

Γk,(p) =

∫(0,∞)

eβAks

∣∣fk(s, Y k,(p)s , Zk,(p)s , Uk,(p)(s, ·)

)∣∣2(αks )2

dCks

(II.16)

≤(II.49)

∫(0,∞)

eβAks (αks )2

∣∣Y k,(p)s

∣∣2 dCks +

∫(0,∞)

eβAks

∣∣cks(Zk,(p)s )∣∣2 +

∣∣∣∣∣∣Uk,(p)s (·)∣∣∣∣∣∣s,k

]dCks

(II.37)

≤(II.49)

C supt∈R+

E[∣∣ξk∣∣2 + Γk,(p−1)

∣∣∣Gkt ]+ C⟨(Zk,(p)s − Zk,(p−1)

s ) ·Xk, +(Uk,(p) − Uk,(p−1)

)? µk

⟩∞.

The analogous bound can be obtained for k =∞. Now we can conclude.


supt∈R+

∣∣Lk,(p)t

∣∣2)k∈N is uniformly integrable.

Proof. Apply Cauchy-Schwartz and use the previous lemma.

IV.5.6. Some comments on the framework of the chapter. In this section we would like tocomment on the nature of the conditions we have imposed to settle the framework of Chapter IV. Let usstart with the first four Conditions (B1) - (B5). We have repeatedly commented that they are inherited


by Theorem III.50. One may question the necessity of the convergence imposed in (B2), which is strongercomparing to the analogous of Theorem III.3, i.e.

Xk,∞ +Xk,

∞L2(G;Rn)−−−−−−→k→∞

X∞,c∞ +X∞,d

∞

instead of

ĎXk∞

L2(G;Rn)−−−−−−→k→∞

ĎX∞∞ .

The main reason is that in Subsection IV.5.4 we need to utilize the convergence(〈Xm,〉, 〈Xm,〉) J1(R××R

×)−−−−−−−−−−→m→∞

(〈X∞,c〉, 〈X∞,d〉) P− a.s.,

a convergence which is not guaranteed by Theorem III.3. On the other hand, Condition (B2) is theone that provides us the flexibility to approximate the continuous part of X∞ with purely discontinuousmartingales, a property which is essential when the discussion comes to numerical schemes.

For the Condition (B6) we have provided some comments at the end if Section IV.1. Therefore, weproceed to Condition (B7).

(B7).(ii) This condition is usually met when we need to guarantee the convergence of compositionsunder the J1−topology, e.g. see Kurtz and Protter [44, Lemma 2.1]. In our case we do not need to bethat abstract since we can exploit the Lipschitz property of the generators. The latter allows us to obtainfor K2 compact subset of R and U ∈ Cc,0(R+ × R) and N ∈ N that, if (yk)k∈N

such that

ykJ1−−→ y∞

then

supt∈[0,N ]

∣∣f∞(ω, λk(t), ykλk(t), z, U(λk(t), ·))− f∞(ω, t, y∞, z, U(t, ·))∣∣, z ∈ K2

−−−−→k→∞

0,

since U is uniformly continuous.(B7).(iii) This condition is a generalisation of the assumption that the generators are globally Lipschitz

which share the same Lipschitz constants. This global property implies in particular that the sequenceof generators forms for every point (y, z, u) a uniformly equicontinuous family of functions. It is thenrelatively straightforward to prove that the convergence for every point (y, z, u) implies the convergence(B7).(iii). For example, when there is no dependence on u, see in the proof of Briand et al. [15,Proposition 11].(B7).(iv) In view of the predictability of the bounds, it is natural to assume the convergence under

the uniform topology. We will comment the assumption of the local boundedness of the Radon–Nikodymderivative when we discuss about Condition (B9).

We proceed now to the discussion about Condition (B9).

(B9).(i) Let us fix a k ∈ N. The analogous arguments hold also for the case k = ∞. Recall thatthe pair (ĎXk, Ck) satisfies Assumption Section I.7. The existence of an integrator Ck is ensured by

Lemma I.166, where we have assumed specific positive functions, say f1, f2, of 〈Xk,〉 and Id ∗ ν(X,Gk)

with the property that . In other words, we have set Ck := f1(〈Xk,〉)+ f2

(Id ∗ ν(X,Gk)

).

On the other hand, in view of the Conditions (B1) - (B4) and Theorem I.160, we know that thesequence (〈Xk,〉, 〈Xk,〉) converges under the J1-topology to (〈X∞,c〉, 〈X∞,d〉). Since we are interestedat this point for absolute continuity of measures, we can choose for every k ∈ N the same positive andcontinuous function f , such that

Ck := f1(〈Xk,〉)+ f2

(〈Xk,〉) for every k ∈ N and C∞ := f1(〈X∞,c〉)+ f2

(〈X∞,d〉).We can, possibly, alternatively choose a sequence (fk)k∈N, which may allow dependence on the time, assoon as it respects the J1−topology.(B9).(ii) This is completely analogous to the previous case once we recall that Theorem I.160 verifies

also the convergence

(〈Xk,〉∞, 〈Xk,〉∞)L2(G;R)−−−−−−→ (〈X∞,c〉∞, 〈X∞,d

∞ 〉).(B9).(iii) Since we have assumed that the bound determined by (II.18) is deterministic and in view

of (B3), this assumption is completely natural.

In view of the comment on the way we propose to choose C∞, the (local) boundedness of the the Radon–Nikodym derivative d〈X∞,c〉/dC∞ is clear.

IV.6. COMPARISON WITH THE RELATED LITERATURE 131

IV.6. Comparison with the related literature

In this section we are going to compare our result with the existing literature. More precisely,the comparison will be made with Madan et al. [50] which is the most general result for discrete-timeapproximations of BSDEs. However, we need to underline that the work of Briand et al. [15] has greatlyinspired the author. In the following we will translate the notation to the one we have used.

We start with the Condition (B1), i.e. the limit-filtration has to be quasi-left-continuous, which iscommon between the two works. Actually, we may recall that Memin [53, Counter-example 3] whichjustifies the necessity of (B1). However, in our case we are not restricted in (square-integrable) Levymartingale, but it can be an arbitrary (square-integrable) one.

We proceed to the conditions imposed on the sequence of driving martingales (ĎXk,, Xk)k∈N. In

[50] the integrators can be multidimensional, while we have presented the real-valued case. However,in view of the results in Chapter III, we can readily adapt the dimension of the integrators to a higherone. The same holds true for the dimension of the solution of the BSDE, in view of the main theorem ofChapter II. In Chapter IV we preferred to present the result in an as simple as possible manner. In bothcases, they are square-integrable martingales whose terminal values converge in mean to the respectiveterminal values of the natural pair of X∞. Now we proceed to describe a few improvements that takeplace under our framework.

• The time horizon in our case can be infinite, while in [50] it is assumed finite.• The terminal values in [50] need to converge in a stronger sense than in L2−mean, see [50, Conditions

(2.2) and (2.5)], while in our case the convergence is in L2−mean. Actually, we have explained at thebeginning of Subsection III.6.1 the role of Lemma III.39 in order to obtain this sharp convergence.

• In our framework we need MμXk,

[ΔXk,∣∣PG

k]= 0 for every k ∈ N, which is a weaker condition

compared to independence assumption imposed in [50]. Let us argue about it. To this end, let us fix afinite time horizon T , a k ∈ N and a partition (tm)m=0,...,M of [0, T ]. We will assume that Xk,, Xk,

have independent increments and are constant on the interval [tm, tm+1) for m = 0, . . . ,M − 1. In thissetting we have for every bounded Gk−predictable function U

MμXk,

[UM

μXk,

[ΔXk,∣∣PG

k]]= M

μXk,

[UΔXk,

]= E

[M−1∑m=0

U(tm,ΔXk,tm )ΔXk,

tm

]= 0.

Finally, regarding the properties the generator should possess, we will restrict ourselves on mentioning thatMadan et al. [50, Assumption 1] demands the generators to be uniformly Lipschitz. In Subsection IV.5.6we have commented that the uniform Lipschitz condition translates the pointwise convergence into seem-ingly stronger kind of convergence. For this reason, we could say that we have used finally the usualconvergence assumptions as they translate in a weaker framework.

APPENDIX A

Auxiliary results

A.1. Auxiliary results of Chapter II

Proof of Lemma II.13. Let (γ, δ) ∈ Cβ . We shall begin by obtaining the critical points of the mapΠΦ. We have

∂

∂γΠΦ(γ, δ) = (2 + 9δ) e(δ−γ)Φ Φγ2 + (2− δΦ)γ − δ

γ2(δ − γ)2,

∂

∂δΠΦ(γ, δ) = − 9

δ2+ e(δ−γ)Φ

[9 + (2 + 9δ)Φ

](δ − γ)

γ(δ − γ)2− 2 + 9δ

γ(δ − γ)2

.

The only possible critical points for ΠΦ are therefore such that δ = −2/9 or γ = δΦ−2±√

4+δ2Φ2

2Φ . However,

the values δ = −2/9 and γ = δΦ−2−√

4+δ2Φ2

2Φ are ruled out as negative. For 0 < δ ≤ β we have(δΦ− 2 +√

4 + δ2Φ2

2Φ, δ)∈ Cβ .

Let us define γΦ(δ) := δΦ−2+√

4+δ2Φ2

2Φ , for 0 < δ ≤ β. It is easy to verify that γΦ(δ) ∈ (0, δ). Then, sometedious calculations yield that

∂ΠΦ

∂δ

(γΦ(δ), δ

)= − 9

δ2−

exp[(δ − γΦ(δ)

)Φ]

γΦ(δ)(δ − γΦ(δ)

)2 · 2γΦ(δ)Φ + 9γΦ(δ) + 2

(γΦ(δ)Φ + 1)2< 0

therefore ΠΦ does not admit any critical point on Cβ , for which 0 < γ < δ < β. Hence, the infimum onthis set is necessarily attained on its boundary. The cases where at least one among δ and γ goes to 0, orwhere their difference goes to 0, lead to the value∞. The only remaining case is therefore 0 < γ < δ = β,where β is fixed. Then we get

d

dγΠΦ(γ, β

)= (2 + 9β) e(β−γ)Φ Φγ2 + (2− βΦ)γ − β

γ2(β − γ)2,

and ΠΦ(γ, β) viewed as a function of γ attains its minimum at its critical point given by γΦ(β), sincedΠΦ

dγ

(γ, β

)< 0 on (0, γΦ(β)) and dΠΦ

dγ

(γ, β

)> 0 on (γΦ(β), β).

Now, we proceed to the second case, and start by determining the critical points of ΠΦ? . It holds

∂

∂γΠΦ? (γ, δ) = − 8

γ2+ 9δe(δ−γ)Φ Φγ2 − (δΦ− 2)γ − δ

γ2(δ − γ)2,

∂

∂δΠΦ? (γ, δ) = − 9

δ2+ 9e(δ−γ)Φ (1 + δΦ)(δ − γ)− δ

γ(δ − γ)2.

Following analogous computations as above we can prove that, for (γ, δ) ∈ Cβ , the equation

∂

∂γΠΦ? (γ, δ) = 0⇔ Pδ(γ) := 8(δ − γ)2 − 9δe(δ−γ)Φ

(Φγ2 − (δΦ− 2)γ − δ

)= 0

has a unique root, say γΦ? (δ), which moreover satisfies γΦ

? (δ) ∈ (γΦ(δ), δ). This can be proved because thefunction Pδ : (0, δ)→ R is decreasing, for each fixed δ ∈ (0, β), with Pδ

(γΦ(δ)

)> 0 and Pδ(δ) < 0. Now

observe that for γ > δ2Φ1+δΦ it holds ∂

∂δΠΦ? (γ, δ) < 0 and that Pδ

(δ2Φ

1+δΦ

)> 0. Using the monotonicity of

Pδ we have that γΦ? (δ) > δ2Φ

1+δΦ and therefore also ∂∂δΠΦ

? (γΦ? (δ), δ) < 0. Arguing as above we can conclude

that the infimum is attained for δ = β at the point(γΦ? (β), β

).

Finally, the limiting statements follow by straightforward but tedious computations.

133

134 A. AUXILIARY RESULTS

A.1.1. The non-exponential submultiplicative case. In this subsection we provide the a prioriestimates of the semi–martingale decomposition (II.20) in the case hζ : R→ [1,∞), where hζ(x) = 1 forx < 0 and hζ(x) := (1 + x)ζ , for x > 0 where ζ is a fixed number greater than 1, with the additionalassumption that the process A defined in (F4) is P−a.s. bounded by Ψ. However, using completelyanalogous arguments the a priori estimates can be obtained for a submultiplicative function h. In [64,Proposition 25.4] (see the proof of (iii)) we can find an easy criterion for an increasing function h : R→ R+

to be submultiplicative. Namely, it suffices for the function h : R → (0,∞) to be flat on (−∞, b] andthe function log(h(·)) to be concave on (b,∞), for some b ∈ R+. It is immediate that the function hζas defined above is submultiplicative. Since ζ is fixed, we will simply denote hζ as h in the rest of thesection.

Before we proceed to the proof we provide the properties of the submultiplicative function we willneed, presented however for the function hζ , as well as some notation. For We start with properties.

(i) h : R+ → [1,∞) and h is non-decreasing.(ii) h is submultiplicative, i.e. there exists ch > 0 such that

h(x+ y) ≤ chh(x)h(y), for all x, y ∈ R+.

(iii) It holds x ≤ h(x) for every x ∈ R+.(iv) The indefinite integral

∫h(s)−γ ds can be explicitly calculated. We define for γ > 1

ζ

Hγ(x) :=

∫(0,x]

h−γ(s) ds =1

1− γζ[(1 + x)1−γζ − 1

]=

1

−γζ + 1

[(1 + x)−γζ+1 − 1

]=

1

−γζ + 1

[(h(x)

)−γζ+1ζ − 1

]=

1

−γζ + 1

[(h(x)

)−γ+ 1ζ − 1

]=

1

−γζ + 1

[hϑ(γ)(x)− 1

], (A.1)

where ϑ(γ) := −γ + 1ζ < 0. Moreover,∫

(t,u]

h−γ(s)ds = Hγ(u)−Hγ(t) = |Hγ(t)| − |Hγ(u)|

=1

| − γζ + 1|[hϑ(γ)(t)− hϑ(γ)(u)] =

1

γζ − 1[hϑ(γ)(t)− hϑ(γ)(u)]. (A.2)

(v) The set ΓH :=γ ∈ R+, Hγ : R+ → (−∞, 0] is increasing

is non–empty. In view of the restriction

in (iv), the set ΓH is indeed non-empty, i.e.

ΓH = (1

ζ,∞). (A.3)

(vi) There exists β ∈ R+ such that

E[ ∫

(0,T ]

hβ(As)‖fs‖22α2s

dCs

]<∞.

By [64, Proposition 25.4] we can calculate the submultiplicative constant

ch = exp[

ln(h(2 · 0)

)− ln

(h(0)

)]= 1.

Lemma A.1. Assume the framework of Lemma II.16 except for the definition of the β−norms, wherewe substitute the exponential function with the function h, for ζ ≥ 1; e.g see (vi). Moreover, assume that

A is bounded by the positive constant Ψ. Then, it holds for 1ζ < γ < δ < β

‖αy‖2H2δ

+ ‖η‖2H2δ≤

[2h(Ψ)hδ(Φ) + 9 +

9[h(Ψ)]1−1ζ [h(Φ)]δ−

1ζ

δ − 1ζ + 1

]‖ξ‖2L2

δ

+

[2[h(Ψ)]1+ 1

ζ [h(Φ)]δ−γ+ 1ζ

δ − γ + 1ζ + 1

+9h(Ψ)[h(Φ)]δ−γ

δ − γ + 1+

9

δζ − 1

]∥∥∥∥fα∥∥∥∥H2δ

,

and

‖y‖2S2δ

+ ‖η‖2H2δ≤[8 + 2h(Ψ)hδ(Φ)

]‖ξ‖2H2

δ

A.1. AUXILIARY RESULTS OF CHAPTER II 135

+

[[h(Ψ)]

1ζ

γζ − 1+

2[h(Ψ)]1+ 1ζ [h(Φ)]δ−γ+ 1

ζ

δ − γ + 1ζ + 1

]∥∥∥∥fα∥∥∥∥2

H2δ

.

Proof. Recall the identity

yt = ξ +

∫(t,T ]

fs dCs −∫

(t,T ]

dηs = E

[ξ +

∫(t,T ]

fs dCs

∣∣∣∣∣Gt], (A.4)

and introduce the anticipating function

Ft =

∫(t,T ]

fs dCs.

For γ ∈ ΓH , we have by the Cauchy-Schwarz inequality,

‖Ft‖22 ≤∫

(t,T ]

h−γ(As) dAs

∫(t,T ]

hγ(As)‖fs‖22α2s

dCs

=

∫(At,AT ]

h−γ(ALs) ds

∫(t,T ]


dCs

Lemma I.34≤(i)

∫(At,AT ]

h−γ(s) ds

∫(t,T ]


dCs

(A.2)=

1

γζ − 1[hϑ(γ)(At)− hϑ(γ)(AT )]

∫(t,T ]


dCs

≤ 1

γζ − 1hϑ(γ)(At)

∫(t,T ]


dCs. (A.5)

For δ ∈ R+ and by integrating (A.5) w.r.t. hδ(At) dAt it follows∫(0,T ]

hδ(At)‖Ft‖22 dAt(A.5)

≤ 1

γζ − 1

∫(0,T ]

hδ+ϑ(γ)(At)

∫(t,T ]


dCs dAt

=1

γζ − 1

∫(0,T ]


∫(0,s)

hδ+ϑ(γ)(At) dAt dCs

≤ 1

γζ − 1

∫(0,T ]


∫(0,s]

hδ+ϑ(γ)(At) dAt dCs

≤ 1

γζ − 1

∫(0,T ]


∫(A0,As]

hδ+ϑ(γ)(ALt) dtdCs, (A.6)

where in the equality we used Tonelli’s Theorem. By [64, Proposition 25.4] we have that hδ+ϑ(γ) re-mains submultiplicative if δ + ϑ(γ) > 0, which is equivalent to δ > |ϑ(γ)|. Therefore, we can applyLemma I.34.(vii) for the function g(x) = hδ+ϑ(γ)(x). Then, Inequality (A.6) becomes∫

(0,T ]

hδ(At)‖Ft‖22 dAtLemma I.34≤ c

δ+ϑ(γ)h hδ+ϑ(γ)(Φ)

∫(0,T ]


∫(A0,As]

hδ+ϑ(γ)(t) dtdCs

ch=1=

hδ+ϑ(γ)(Φ)

δ + ϑ(γ) + 1

∫(0,T ]


(hδ+ϑ(γ)+1(As)− hδ+ϑ(γ)+1(A0)

)dCs

A0=0≤

h(A0)=1

hδ+ϑ(γ)(Φ)

δ + ϑ(γ) + 1

∫(0,T ]

[h(As)

]δ+ 1ζ+1 ‖fs‖22

α2s

dCs, (A.7)

which is integrable if δ + 1ζ + 1 ≤ β; see (vi). Define

% := δ +1

ζ+ 1 and Λγ,δ,Φ :=

hδ+ϑ(γ)(Φ)

δ + ϑ(γ) + 1.

To sum up, for

γ ∈ ΓH , δ > γ − 1

ζand 0 < % ≤ β, (A.8)


we have

E[ ∫

(0,T ]

hδ(At)‖Ft‖22dAt

]≤ Λγ,δ,Φ

∥∥∥fα

∥∥∥2

H2%

. (A.9)

For the estimate of ‖αy‖H2δ

we first use the fact that

‖αy‖2H2δ

= E[ ∫

(0,T ]

hδ(At)‖yt‖22 dAt

]≤ 2E

[ ∫(0,T ]

E[hδ(At)‖ξ‖22 + hδ(At)‖Ft‖22

∣∣Gt]dAt

]= 2E

[ ∫(0,∞)

E[hδ(At)‖ξ‖22 + hδ(At)‖Ft‖22

∣∣Gt] dATt

]=2E

[ ∫(0,∞)

hδ(At)‖ξ‖22 + hδ(At)‖Ft‖22 dATt

]= 2E

[ ∫(0,T ]

hδ(At)‖ξ‖22 dAt

]+ 2E

[ ∫(0,T ]

hδ(At)‖Ft‖22 dAt

]= 2E

[ ∫(A0,AT ]

hδ(ALs)‖ξ‖22 ds]

+ 2E[ ∫

(0,T ]


]= 2cδhh

δ(Φ)E[‖ξ‖22

∫(A0,AT ]

hδ(s) ds]

+ 2E[ ∫

(0,T ]


]=ch=1

2hδ(Φ)E[‖ξ‖22hδ(λ)(AT −A0)

]+ 2E

[ ∫(0,T ]


], for λ ∈ [A0, AT ] or

2δζ+1h

δ(Φ)E[‖ξ‖22

(hδ+

1ζ (AT )− hδ+

1ζ (A0)

)]+ 2E

[ ∫(0,T ]


](A.10)

(i),(iii)

≤A0=0,(A.9)

2hδ(Φ)‖ξ‖2L2

δ+1+ 2Λγ,δ,Φ,ch

∥∥∥ fα∥∥∥2

H2%

.

2δζ+1h

δ(Φ)‖ξ‖2L2

δ+ 1ζ

+ 2Λγ,δ,Φ,ch∥∥∥ fα∥∥∥2

H2%

.(A.11)

In the second equality we have used that the processes ‖ξ‖221Ω×[0,∞](·) and |F (·)|2 are uniformly in-tegrable, hence their optional projections are well defined. Indeed, using (A.5) and recalling thatE[‖F0‖22] <∞, we can conclude the uniform integrability of |F (·)|2. Then, it holds that

ΠGo

(hδ(A·)‖ξ‖22 + hδ(A·)|F (·)|2

)t

= hδ(At)ΠGo

(‖ξ‖221Ω×[0,∞](·)

)t

+ hδ(At)ΠGo

(|F (·)|2

)t

= hδ(At)E[‖ξ‖22

∣∣Gt]+ hδ(At)E[‖Ft‖22

∣∣Gt]= E

[hδ(At)‖ξ‖22 + hδ(At)‖Ft‖22

∣∣Gt] ,which justifies the use of Corollary I.23.

For the estimate of ‖y‖S2δ

we have

‖y‖S2δ

= E[

sup0≤t≤T

(hδ2 (At)‖yt‖2

)2] ≤ E[

sup0≤t≤T

(hδ2 (At)E

[‖ξ‖2 + ‖Ft‖2

∣∣Gt])2]= E

[sup

0≤t≤TE[hδ2 (At)‖ξ‖2 + h

δ2 (At)‖Ft‖2

∣∣Gt]2] ≤ 2E

[sup

0≤t≤TE[√

hδ(At)‖ξ‖22 + hδ(At)‖Ft‖22

∣∣∣∣Gt]2]

(A.5)

≤ 2E

sup0≤t≤T

E

[√hδ(AT )‖ξ‖22 +

1

γζ − 1hδ+ϑ(γ)(At)

∫(t,T ]


dCs

∣∣∣∣∣Gt]2

≤ 2E

sup0≤t≤T

E

[√hδ(AT )‖ξ‖22 +

1

γζ − 1

∫(t,T ]

[h(As)]γ+δ+ϑ(γ)‖fs‖22α2s

dCs

∣∣∣∣∣Gt]2

≤ 2E

sup0≤t≤T

E

[√hδ(AT )‖ξ‖22 +

1

γζ − 1

∫(0,T ]

[h(As)]δ+ 1

ζ‖fs‖22α2s

dCs

∣∣∣∣∣Gt]2

A.1. AUXILIARY RESULTS OF CHAPTER II 137

≤ 8E

[hδ(AT )‖ξ‖22 +

1

γζ − 1

∫(0,T ]

[h(As)]δ+ 1

ζ‖fs‖22α2s

dCs

]

= 8‖ξ‖2H2δ

+1

γζ − 1

∥∥∥∥fα∥∥∥∥2

H2

δ+ 1ζ

. (A.12)

where in the second and third inequalities we used the inequality a+ b ≤√

2(a2 + b2) and (A.5) respec-tively.

What remains is to control ‖η‖H2δ. We remind once more the reader that∫ T

t

dηs = ξ − yt + Ft,

hence

E[‖ξ − yt − Ft‖22

∣∣Gt] = E

[∫(t,T ]

dTr[ 〈η〉s]

∣∣∣∣∣Gt]. (A.13)

In addition, we have

hδ(As) = (1 +As)δζ = δζ

∫(A0,As]

(1 + x)δζ−1dx+ 1 = δζ

∫(A0,As]

hδ−1ζ (x)dx+ 1 (A.14)

∫(0,T ]

hδ(As) dTr[ 〈η〉s](A.14)

≤ δζ

∫(0,T ]

∫(A0,As]

hδ−1ζ (t) dt dTr[ 〈η〉s] + Tr[ 〈η〉T ]

Lemma I.34≤(i)

δζ

∫(0,T ]

∫(A0,As]

hδ−1ζ (ALt) dt dTr[ 〈η〉s] + Tr[ 〈η〉T ]

= δζ

∫(0,T ]

∫(0,s]

hδ−1ζ (At) dAtdTr[ 〈η〉s] + Tr[ 〈η〉T ]

≤ δζ

∫(0,T ]

hδ−1ζ (At)

∫(t,T ]

dTr[ 〈η〉s]dAt + Tr[ 〈η〉T ],

so that

‖η‖H2δ≤ δζE

[∫(0,T ]

hδ−1ζ (At)

∫(t,T ]

dTr[ 〈η〉s] dAt

]+ E [Tr[ 〈η〉T ]] . (A.15)

We now need an estimate for E[∫

(0,T ]dTr[ 〈η〉s]

], which is given by

E [Tr[ 〈η〉T ]] = E[|ξ − y0 + F (0)|2

]≤ 3E

[‖ξ‖22 + ‖y0‖22 + ‖F0‖22

](A.4)

≤ 9E[‖ξ‖22

]+ 9E

[‖F0‖22

] (A.5)

≤(i)

9‖ξ‖L2δ

+ 91

δζ − 1

∥∥∥∥fα∥∥∥∥H2δ

for δ ∈ ΓH

(A.5)

≤ 9‖ξ‖L2δ

+9

δζ − 1

∥∥∥∥fα∥∥∥∥H2δ

, (A.16)

where we used the fact that E[‖y0‖22

]≤ 2E

[‖ξ‖22 + ‖F0‖22

].

For the first summand on the right-hand-side of (A.15), we have

E

[∫(0,T ]

hδ−1ζ (At)

∫(t,T ]

dTr[ 〈η〉s] dAt

]Lemma I.23

= E

[∫(0,T ]

hδ−1ζ (At)E

[∫(t,T ]

dTr[ 〈η〉s]

∣∣∣∣∣Gt]

dAt

](A.13)

= E

[∫(0,T ]

hδ−1ζ (At)E

[|ξ − yt + Ft|2

∣∣Gt] dAt

]

≤ 3E

[∫(0,T ]

hδ−1ζ (At)E

[‖ξ‖22 + ‖yt‖22 + ‖Ft‖22

∣∣Gt] dAt

](A.4)

≤ 3E

[∫(0,T ]

hδ−1ζ (At)‖ξ‖22 dAt

]+ 3E

[∫(0,T ]

hδ−1ζ (At)‖Ft‖22 dAt

]


+ 6E

[∫(0,T ]

hδ−1ζ (At)E

[‖ξ‖22 + ‖Ft‖22

∣∣Gt] dAt

]Lem. I.23

= 9E

[∫(0,T ]

hδ−1ζ (At)‖ξ‖22 dAt

]+ 9E

[∫(0,T ]


](γ∈ΓH)

= 9E

[∫(A0,AT ]

hδ−1ζ (ALt)‖ξ‖22 dt

]+ 9E

[∫(0,T ]


]Cor. I.36≤

(A.9)9hδ−

1ζ (Φ)E

[‖ξ‖22

∫(A0,AT ]

hδ−1ζ (t) dt

]+ 9Λγ,δ−

1ζ ,Φ

∥∥∥∥fα∥∥∥∥H2δ+1

for δ + 1 ≤ β

≤ 9hδ−1ζ (Φ)

δ − 1ζ + 1

E[‖ξ‖22

[hδ−

1ζ+1(AT )− 1

]dt]

+ 9Λγ,δ−1ζ ,Φ

∥∥∥∥fα∥∥∥∥H2δ+1

≤ 9hδ−1ζ (Φ)

δ − 1ζ + 1

‖ξ‖δ− 1ζ+1 + 9Λγ,δ−

1ζ ,Φ

∥∥∥∥fα∥∥∥∥H2δ+1

. (A.17)

In total, Inequality (A.15) can be written

‖η‖H2δ≤ δζE

[∫ T

0

hδ−1ζ (At)

∫(t,T ]

dTr[ 〈η〉s] dAt

]+ E [Tr[ 〈η〉T ]]

(A.16)

≤(A.17)

9hδ−1ζ (Φ)

δ − 1ζ + 1

‖ξ‖δ− 1ζ+1 + 9Λγ,δ−

1ζ ,Φ

∥∥∥∥fα∥∥∥∥H2δ+1

+ 9‖ξ‖L2δ

+9

δζ − 1

∥∥∥∥fα∥∥∥∥H2δ

(A.18)

In order to be able to construct a contraction, we need to have on the right hand side of (A.11), resp.

(A.12), and (A.18) only ‖ fα‖H2δ. One way to overcome this problem is to assume that A is P − a.s.

bounded, say by Ψ. Then, the Inequality (A.11) can be written as

‖αy‖2H2δ≤

2hδ(Φ)‖ξ‖2L2

δ+1+ 2Λγ,δ,Φ,ch

∥∥∥ fα∥∥∥2

H2%

2δζ+1h

δ(Φ)‖ξ‖2L2

δ+ 1ζ

+ 2Λγ,δ,Φ,ch∥∥∥ fα∥∥∥2

H2%

≤

2h(Ψ)hδ(Φ)‖ξ‖2L2

δ+ 2[h(Ψ)]

1+ 1ζ [h(Φ)]δ+ϑ(γ)

δ+ϑ(γ)+1

∥∥∥ fα∥∥∥H2δ

,

2δζ+1h

1ζ (Ψ)hδ(Φ)‖ξ‖2L2

δ+ 2[h(Ψ)]

1+ 1ζ [h(Φ)]δ+ϑ(γ)

δ+ϑ(γ)+1

∥∥∥ fα∥∥∥H2δ

.(A.19)

the Inequality (A.12) can be written as

‖y‖S2δ≤ 8‖ξ‖2H2

δ+

1

γζ − 1

∥∥∥∥fα∥∥∥∥2

H2

δ+ 1ζ

≤ 8‖ξ‖2H2δ

+[h(Ψ)]

1ζ

γζ − 1

∥∥∥∥fα∥∥∥∥2

H2δ

(A.20)

and the Inequality (A.18) can be written as

‖η‖H2δ≤

(9 +

9[h(Ψ)]1−1ζ [h(Φ)]δ−

1ζ

δ − 1ζ + 1

)‖ξ‖δ +

(9h(Ψ)Λγ,δ−

1ζ ,Φ +

9

δζ − 1

)∥∥∥fα

∥∥∥2

H2δ

=

(9 +

9[h(Ψ)]1−1ζ [h(Φ)]δ−

1ζ

δ − 1ζ + 1

)‖ξ‖δ +

(9h(Ψ)[h(Φ)]δ−γ

δ − γ + 1+

9

δζ − 1

)∥∥∥fα

∥∥∥2

H2δ

. (A.21)

A.2. Auxiliary results of Chapter III

Lemma A.2. Let (Lk)k∈N be a sequence of Rp−valued processes and (Nk)k∈N be a sequence of Rq−valuedprocesses such that

(i)(Tr[[Lk]∞

])k∈N is uniformly integrable and

(ii)(Tr[[Nk]∞

])k∈N is bounded in L1(G;R), i.e. sup

k∈NE[Tr[[Nk]∞

]]<∞.

A.2. AUXILIARY RESULTS OF CHAPTER III 139

Then(∥∥Var

([Lk, Nk]

)∞

∥∥1


Proof. By Corollary I.29.(iv), it is sufficient to prove that the sequence(Var([Lk,i, Nk,j ]

)∞

)k∈N

is uniformly integrable, for every i = 1, . . . , p and j = 1, . . . , q. We will use Theorem I.25 in orderto prove the required property. Let i, j be arbitrary but fixed. The L1–boundedness of the sequence(Var([Lk,i, Nk,j ])∞)k∈N is obtained by means of the Kunita–Watanabe inequality in the form I.26 andthe L1−boundedness of the sequences

([Lk,i]∞

)k∈N and

([Nk,j ]∞

)k∈N; the former is L1−bounded as

uniformly integrable. By the Kunita–Watanabe inequality again, but now in the form (I.24), and theCauchy–Schwarz inequality, we obtain for any A ∈ G∫

A

Var([Lk,i, Nk,j ]

)∞ dP

K-W ineq.

≤∫A

[Lk,i]12∞[Nk,j ]

12∞dP

C-S ineq.

≤(∫

A

[Lk,i]∞ dP) 1

2(∫

A

[Nk,j ]∞ dP) 1

2

≤(∫

A

[Lk,i]∞ dP) 1

2(

supk∈N

∫Ω

[Nk,j ]∞ dP) 1

2

≤ C(∫

A

[Lk,i]∞ dP) 1

2

, (A.22)

where C2 := supk∈N

E[Tr[[Nk]∞

]]. Note that for the third inequality we used that [Nk,j ]∞ ≥ 0 for all

k ∈ N.

Now we use [35, Theorem 1.9] for the uniformly integrable sequence([Lk,i]∞

)k∈N. For every ε > 0,

there exists δ > 0 such that, whenever P(A) < δ, it holds

supk∈N

∫A

[Lk,i]∞ dP <ε2

C2⇐⇒ sup

k∈N

(∫A

[Lk,i]∞ dP)1/2

<ε

C.

This further implies supk∈N

E[1AVar

([Lk,i, Nk,j ]∞

)] (A.22)< ε, which is the required condition.

The proof for the predictable quadratic variation is completely analogous. We state the result,however, for our convenience.

Lemma A.3. Let (Lk)k∈N be a sequence of Rp−valued processes and (Nk)k∈N be a sequence of Rq−valuedprocesses such that

(i) 〈Lk〉 and 〈Nk〉 are well-defined for every k ∈ N,(ii)

(Tr[〈Lk〉∞

])k∈N is uniformly integrable and

(iii)(Tr[〈Nk〉∞

])k∈N is bounded in L1(G;R), i.e. sup

k∈NE[Tr[[Nk]∞

]]<∞.

Then(∥∥Var

(〈Lk, Nk〉

)∞

∥∥1


In the following we provide some results which can be seen as simple exercises in measure theory.

Lemma A.4. Let D be a countable and dense subset of R. Then every open U ⊂ R is the countable unionof intervals with endpoints in D, i.e. there exists a sequence of intervals

((ak, bk)

)k∈N with ak, bk ∈ D,

for every k ∈ N, such that U = ∪k∈N(ak, bk).

Proof. Let U be an open subset of R. For every x ∈ U there exists an rx > 0 such that B(x, rx) ⊂ U ,where B(y, r) = z ∈ R, |y−z| < r. Due to the density of D we can choose ax, bx ∈ D such that ax < bxand (ax, bx) ⊂ B(x, rx) ⊂ U. Then clearly ∪x∈U (ax, bx) ⊂ U. On the other hand, D is countable andtherefore D×D is also countable. Moreover, we can naturally associate to each ordered pair a, a, bthe interval (a, b) if a < b or the empty set if a ≥ b, which proves that there are at most countably manyintervals with endpoints in D.

Corollary A.5. Let U ⊂ R \ 0 be open. Then for every i = 1, . . . , ` there exists a countable subfamilyof I(X∞,i), which has been introduced in Section III.4, name (U i,k)k∈N, such that U = ∪k∈NU i,k.

Proof. Let U be an open subset of R \ 0 and i ∈ 1, . . . , `. Then, there exist open sets U+, U−

such that U+ ⊂ (0,∞), U− ⊂ (−∞, 0) and U = U+∪U−. On the other hand, by [41, Lemma VI.3.12] wecan conclude that the set I(X∞,i) is at most countable. Therefore, the complement of I(X∞,i), denoteit by I(X∞,i)c, is dense in R. Indeed, if I(X∞,i)c was not dense, there would exist an open subset V ofR such that I(X∞,i)c ∩ V = ∅ or equivalently V ⊂ I(X∞,i). However this is a contradiction, since V isuncountable and I(X∞,i) is countable.


The above allow us to apply Lemma A.4 for D = I(X∞,i) in order to find two countable families(U+,i,k)k∈N and (U−,i,k)k∈N such that

U+ =⋃k∈N

U+,i,k and U− =⋃k∈N

U−,i,k.

The required countable family is (U+,i,k)k∈N ∪ (U−,i,k)k∈N.

Lemma A.6. It holds σ(J (X∞)

)= B(R`), where J (X∞) has been introduced in Subsection III.4 and

σ(J (X∞)

)denotes the σ−algebra in R` generated by the family J (X∞).

Proof. The space R` is a finite product of the second countable metric space R. Therefore, it holds

B(R`) = B(R)⊗ · · · ⊗ B(R)︸︷︷︸`−times

and to this end, it is sufficient to prove that σ(I(X∞,i)) = B(R) for every i = 1, . . . , `.

By Corollary A.5 we have

σ(I(X∞,i)) = σ(U ⊂ R \ 0, U is open) = σ(U ⊂ R \ 0, U is open ∪ R ∪ 0)= σ(U ⊂ R, U is open) = B(R),

which allows us to conclude.

Lemma A.7. Let (Σ,S) be a measurable space and %1, %2 be two finite signed measures on this space.Let A be a family of sets with the following properties:

(i) A is a π−system, i.e. if A,B ∈ A, then A ∩B ∈ A.(ii) It holds σ(A) = S

(iii) %1(A) = %2(A) for every A ∈ A.Then it holds %1 = %2.

Proof. Let %+i , %

−i be the (positive) measures obtained by the Jordan decomposition of %i, for

i = 1, 2. By (iii) we obtain

%+1 (A)− %−1 (A) = %+

2 (A)− %−2 (A), for every A ∈ A.However, the above equality can be translated into equality of measures, i.e.

%+1 (A) + %−2 (A) = %+

2 (A) + %−1 (A), for every A ∈ A.The class

C = A ∈ S, %+1 (A) + %−2 (A) = %+

2 (A) + %−1 (A)

can be proven to be a λ−system.1 Now we can conclude that the (positive) measures %+1 +%−2 and %+

2 +%−1coincide on D(A) ⊂ C, where we have denoted by D(A) the λ−system generated by A. Using now theproperty (i) and the π − λ Lemma,2 see [4, Lemma 4.11], we can conclude that the (positive) measures%+

1 + %−2 and %+2 + %−1 coincide on σ(A) = S.

Let, now, P1, N1, resp. P2, N2, the Hahn decomposition of the signed measure %1, resp. %2, wherewith Pi we denote the set of positive mass of %i and with Ni we denote the set of negative mass of %i,for i = 1, 2. In view of the %+

i ⊥ %−i , for i = 1, 2 and of the following equalities

Ω = (P1 ∩ P2) ∪ (P1 ∩N2) ∪ (N1 ∩ P2) ∪ (N1 ∩N2)

%+1 (P1 ∩ P2) = %+

1 (P1 ∩ P2) + %−2 (P1 ∩ P2) = %+2 (P1 ∩ P2) + %−1 (P1 ∩ P2) = %+

2 (P1 ∩ P2)

%+1 (P1 ∩N2) + %−2 (P1 ∩N2) = %+

2 (P1 ∩N2) + %−1 (P1 ∩N2) = 0

0 = %+1 (N1 ∩ P2) + %−2 (N1 ∩ P2) = %+

2 (N1 ∩ P2) + %−1 (N1 ∩ P2)

%−2 (N1 ∩N2) = %+1 (N1 ∩N2) + %−2 (N1 ∩N2) = %+

2 (N1 ∩N2) + %−1 (N1 ∩N2) = %−1 (N1 ∩N2)

we can conclude that P1 ∩ P2, N1 ∩N2 is a common Hahn decomposition of the signed measures %1 and%2. Therefore

1A λ−system is also referred to as Dynkin system or d−system or Sierpinski class. In [4, Foonote 4, p. 135] theinterested reader can find references for this ambiguity in terminology. In [10] is used the term σ−additive class.

2Also known as Dynkin’s Lemma.

A.3. AUXILIARY RESULTS OF CHAPTER IV 141

• For every A ∈ B ⊂ (P1 ∩ P2), B ∈ S we obtain %+1 (A) = %+

2 (A), hence we can conclude that%+

1 = %+2 .

• For every A ∈ B ⊂ (N1 ∩ N2), B ∈ S we obtain %−1 (A) = %−2 (A), hence we can conclude that%−1 = %−2 .

A.3. Auxiliary results of Chapter IV

A.3.1. Moore–Osgood Theorem. The Moore–Osgood Theorem, whose well-known form is The-orem A.8, provides sufficient conditions for the existence of the limit of a doubly-indexed sequence andit can be seen as a special case of Rudin [63, Theorem 7.11]. Here we provide a second form, since thesecond time3 we need to apply the aforementioned theorem we need to relax the existence of the pointwiselimits; see Condition (ii) of Theorem A.8. For more comments on the existence of the iterated limits andof the joint limit of a doubly-indexed sequence the interested reader should consult Hobson [37, ChapterVI, Sections 336-338]. Specifically for the validity of the Theorem A.9 see [37, Chapter VI, Section 337,p. 466] and the reference therein.

Theorem A.8. Let (Γ, dΓ) be a metric space and (γk,p)k,p∈N be a sequence. If

(i) limp→∞

supk∈N

dΓ(γk,p, γk,∞) = 0 and

(ii) limk→∞

dΓ(γk,p, γ∞,p) = 0 for all p ∈ N,

then the joint limit limk,p→∞

γn,k exists. In particular holds limk,p→∞

γk,p = limp→∞

γ∞,p = limk→∞

γk,∞.

Theorem A.9. Let (R, | · |) and (ϑk,p)k,p∈N be a sequence. If

(i) limp→∞

supk∈N|ϑk,p − ϑk,∞| = 0 and

(ii) limp→∞

(lim supk→∞

ϑk,p − lim infk→∞

ϑk,p)

= 0,

then the joint limit limk,p→∞

ϑk,p exists.

A.3.2. Weak convergence of measures on the positive real line.

Definition A.10. Let (µk)k∈N be a countable family of measures on(R+,B(R+)

). We will say that the

sequence (µk)k∈N converges weakly to the measure µ∞ if for every f :(R+, ‖ · ‖∞

)→(R, | · |

)continuous

and bounded holds

limk→∞

∣∣∣∣∫[0,∞)

f(x)µk(dx)−∫

[0,∞)

f(x)µ∞(dx)

∣∣∣∣ = 0.

We denote the weak convergence of (µk)k∈N to µ∞ by µkw−−→ µ∞.

In this section we will use the set

W+0,∞ :=

C ∈ D(R)

∣∣ C is increasing, with C0 = 0 and limt→∞

Ct ∈ R.

Remark A.11. We can extend every element of W+0,∞, name C its arbitrary element, on [0,∞] such

that it is left-continuous at the symbol ∞ by defining C∞ := limt→∞ Ct.

We provide in the following proposition some convenient equivalence for the weak convergence offinite measures. The statement is tailor-made to our later needs, but the interested reader may consultBogachev [10, Section 8.1 - Section 8.3]. Then we provide in Theorem A.14 a new, to the best knowledgeof the author, characterisation of weak convergence of finite measures to an atomless measure definedon the positive real line, which uses relatively compact sets of the Skorokhod space instead of relativelycompact sets of the space of continuous functions defined on R+ endowed with the ‖ · ‖∞−topology.

Proposition A.12. Let (µk)k∈N be sequence of finite measures on(R+,B(R+)

), and denote the as-

sociated distribution functions by Ck, for k ∈ N, for which the finiteness of the measure translates toCk∞ := limt→∞ Ckt ∈ R+ for every k ∈ N. We assume the following

(i) The sequence (µk)k∈N is bounded, i.e. supk∈N µk(R+) <∞. Equivalently, supk∈N Ck∞ <∞.

3The first one is in the proof of Theorem IV.10, while the second is in the proof of Proposition A.16


(ii) The sequence (µk)k∈N is tight. In other words, for every ε > 0 there exists a compact set I suchthat supk∈N µk(R+ \ I) < ε.4 Equivalently, supk∈N |Ck∞ − CkN | < ε for some N ∈ R+.

(iii) µk(0) = 0 for every k ∈ N. Equivalently, Ck ∈ W+0,∞ for every k ∈ N.

(iv) µ∞ is atomless, i.e. µ∞(t) = 0 for every t ∈ R+. Equivalently, C∞ is continuous.

The following are equivalent:

(I) µkw−−→ µ.

(II) Ckt −−−−→k→∞

C∞t , for every t ∈ R+.

(III) CkJ1(R)−−−−→ C∞.

(IV) Cklu−−→ C∞.

(V) supI∈IN

|µk(I)− µ∞(I)| −−−−→k→∞

0 for every N ∈ N, where IN :=I ⊂ [0, N ] | I is interval

.

Proof. The equivalence between (I) and (II) is a classical result, e.g. see Bogachev [10, Proposition8.1.8]. The equivalences between (II),(III) and (IV) are provided by Jacod and Shiryaev [41, TheoremVI.2.15.c.(i), Proposition VI.1.17 b)]. We obtain the equivalence between (IV) and (V) in view of thevalidity of the following inequalities for every N ∈ N:

supt∈[0,N ]

|Ckt − C∞t | ≤ supI∈IN

|µk(I)− µ∞(I)| (because Ck0 = C∞0 = 0)

≤ sups,t∈[0,N ]

|Ckt− − Cks − C∞t + C∞s |+ sups,t∈[0,N ]

|Ckt − Cks − C∞t + C∞s |

+ sups,t∈[0,N ]

|Ckt− − Cks− − C∞t− + C∞s−|

≤ 3 supt∈[0,N ]

|Ckt − C∞t |+ 2 supt∈[0,N ]

|Ckt− − C∞t−|+ supt∈[0,N ]

|Ckt− − C∞t |

≤ 5 supt∈[0,N ]

|Ckt − C∞t |+ supt∈[0,N ]

|Ckt− − Ckt |+ supt∈[0,N ]

|Ckt − C∞t | (Ckis cadlag for every k ∈ N)

≤ 6 supt∈[0,N ]

|Ckt − C∞t |+ supt∈[0,N ]

|Ckt− − Ckt |.

Then Jacod and Shiryaev [41, Lemma VI.2.5] implies that

lim supk→∞

supI∈IN

|µk(I)− µ∞(I)| ≤ 6 limk→∞

supt∈[0,N ]

|Ckt − C∞t |+ lim supk→∞

supt∈[0,N ]

|Ckt− − Ckt |

≤ supt∈[0,N ]

|C∞t− − C∞t | = 0.

The following lemma provides a useful criterion to conclude the equivalence between δ‖·‖∞ -convergence

and δJ1(R+)-convergence in D

([0,∞]

).

Lemma A.13. Let (Ck)k∈N ⊂ W+0,∞ which satisfies the following properties:

(i) CkJ1(R)−−−−→k→∞

C∞.

(ii) Ck∞|·|−−−−→

k→∞C∞∞ .

(iii) The limit function C∞ is continuous.

Then it holds Ck‖·‖∞−−−−−→k→∞

C.

Proof. Let us fix an arbitrary ε > 0. We start by exploiting the fact that C∞ ∈ W+0,∞. Then, there

exists K > 0 such that

supt≥K

(C∞∞ − C∞t ) <ε

4. (A.23)

By (ii), there exists k1 = k1(ε) ∈ N such that

supk≥k1

|Ck∞ − C∞∞ | <ε

4. (A.24)

4Without loss of generality we can assume that I is of the form [0, N ], for some N > 0.


By (iii) there exists s > K such that Cs = 12 (CK + C∞∞ ). By (i), (iii) and Proposition I.113, there exists

k2 = k2(ε, s) such that

supk≥k2

sup0≤t≤s

|Ckt − C∞t | <1

4(C∞∞ − CK) <

ε

16, (A.25)

where the last inequality is valid in view of (A.23), since s > K. Using the fact that the functions areincreasing we derive

supk≥k1∨k2

(sup

s≤t<∞Ck∞ − Ckt

)≤ supk≥k1∨k2

(Ck∞ − Cks

)≤ sup

k≥k1∨k2

∣∣Ck∞ − C∞∞ ∣∣+ supk≥k1∨k2

∣∣C∞∞ − C∞s ∣∣+ supk≥k1∨k2

∣∣C∞s − Cks ∣∣(A.24)<

(A.25)

ε

4+ε

4+

ε

16<ε

2. (A.26)

Therefore, by combining the above, we obtain for every k ≥ k1 ∨ k2

supt∈R+

|Ckt − C∞t | ≤ sup0≤t≤s

|Ckt − C∞t |+ sups≤t≤∞

|Ckt − C∞t |(A.25)<

(A.26)< ε.

Recall that the Kolmogorov metric δKolm(C1, C2) := ‖C1 − C2‖∞, for C1, C2 ∈ W+0,∞, dominates

the Levy metric, which metrises the weak topology on the space of finite measures. Hence the previous

lemma can be seen as a criterion for the weak convergence µCkw−−→ µC∞ , where µCk , for k ∈ N is the

finite measures associated to the increasing function Ck ∈ W+0,∞.

The next step is to provide a characterisation of the aforementioned weak convergence of measuresusing relatively compact sets of D. Since, however, a relatively compact set of D is, in general, onlylocally uniformly bounded, see Theorem I.111, we have to restrict ourselves to those relatively compactsubsets which are uniformly bounded, so that the integrals make sense. The following theorem can beregarded as an extension of Parthasarathy [56, Theorem 6.8] in the special case that the limiting measureis an atomless measure on R+. For α ∈ D(Rp) the modulus w′N has been defined in Definition I.108.

Theorem A.14. Let the measurable space(R+,B(R+)

), (µk)k∈N be a sequence of finite measures such

that µ∞ is an atomless finite measure and A ⊂ D(R). Assume that the following conditions are true

(i) µkw−−→ µ∞.

(ii) A is uniformly bounded, i.e. supα∈A ‖α‖∞ <∞ and(iii) A is relatively compact. In other words, lim

ζ↓0supα∈A

w′N (α, ζ) = 0 for every N ∈ N.

Then,

limk→∞

supα∈A

∣∣∣∣∫[0,∞)

α(x)µk(dx)−∫

[0,∞)

α(x)µ∞(dx)

∣∣∣∣ = 0. (A.27)

Proof. We will decompose initially the quantity whose convergence we intent to prove as follows

supα∈A

∣∣∣∣∫[0,∞)

α(x)µk(dx)−∫

[0,∞)

α(x)µ∞(dx)

∣∣∣∣≤ supα∈A

∣∣∣∣∫[0,N ]

α(x)µk(dx)−∫

[0,N ]

α(x)µ∞(dx)

∣∣∣∣+ supα∈A

∣∣∣∣∫[N,∞)

α(x)µk(dx)


∣∣∣∣∫[N,∞)

α(x)µ∞(dx)

∣∣∣∣ (A.29)

for some N > 0 to be determined and then the first summand of the right-hand side will be decomposedas follows

supα∈A

∣∣∣∣∫[0,N ]

α(x)µk(dx)−∫

[0,N ]

α(x)µ∞(dx)

∣∣∣∣


≤ supα∈A

∣∣∣∣∫[0,N ]

α(x)µk(dx)−∫

[0,N ]

α(x) µαk (dx)


∣∣∣∣∫[0,N ]

α(x) µαk (dx)−∫

[0,N ]

α(x) µα∞(dx)


∣∣∣∣∫[0,N ]

α(x) µα∞(dx)−∫

[0,N ]

α(x)µ∞(dx)

∣∣∣∣(A.31)

where the measures µαk , for k ∈ N and α ∈ A, will be constructed given the measure µk and the functionα. For the second and third summand of (A.29) we will prove that they become arbitrarily small forlarge N > 0. Then, we will conclude once we obtain the convergence to 0 as k →∞ of the summands of(A.31). To this end, let us fix an ε > 0.

Condition (i) implies that supk∈N µk(R+) < ∞as well as the tightness of the sequence by theconvergence µk(R+) −−−−→

k→∞µ∞(R+). Hence, for M := supα∈A ‖α‖∞ <∞, which is Condition (ii), there

exists N = N(ε,M) > 0 such that

supk∈N

µk([N,∞)

)<

ε

2M, which implies sup

k∈N,α∈A

∣∣∣∣ ∫[N,∞)

α(x)µk(dx)

∣∣∣∣ < ε

2.

In other words, we have proven that the second and third summand of the right-hand side of (A.29)become arbitrarily small for large N. In the following N is assumed fixed, but large enough so that theabove hold.

We proceed now to construct for every k ∈ N and α ∈ A a suitable measure µαk on(R+,B(R+)

)such

that we can obtain the convergence of the summands in (A.31). Define KN := supk∈N µk ([0, N ]) < ∞.By (iii) there exists ζ = ζ(ε,KN ) > 0 such that for every ζ ′ < ζ it holds supα∈Aw

′N (α, ζ ′) < ε

4KN. By

the definition of w′ (see Definition I.108) for every α ∈ A and every ζ ′ < ζ, there exists a ζ ′-sparse set

PN,ζ′

α , i.e.

PN,ζ′

α :=

0 = t(0;PN,ζ′

α ) < t(1;PN,ζ′

α ) < . . . < t(κα;PN,ζ′

α ) = N :

mini=1,...,κα−1

(t(i;PN,ζ

′

α )− t(i− 1;PN,ζ′

α ))> ζ ′

,

such that

maxi=1,...,κα

sup|α(s)− α(u)| : s, u ∈

[t(i− 1;PN,ζ

′

α ), t(i;PN,ζ′

α ))

< w′N (α, ζ ′) +ε

4KN

< supα∈A

w′N (α, ζ ′) +ε

4KN<

ε

2KN.

(A.32)

To sum up, for every (α, ζ ′) ∈ A × (0, ζ) we can find a partition of [0, N ], PN,ζ′

α , satisfying (A.32). For

the following, given a finite measure on(R+,B(R+)

), name λ, and a sparse set PN,ζ

′

α

. we will denote A(i;PN,ζ′

α ) := [t(i− 1;PN,ζ′

α ), t(i;PN,ζ′

α )) for i = 1, . . . , κα and

. we define the finite measure λα :(R+,B(R+)

)→ [0,∞) by

λα(·) :=

κα∑i=1

λ(A(i;PN,ζ

′

α ))δt(i−1;PN,ζ

′α )

(·),

where δx is the Dirac measure sitting at the point x. With the above notation we have∣∣∣∣∣∫

[0,N ]

α(x)λ(dx)−∫

[0,N ]

α(x) λα(dx)

∣∣∣∣∣ =

∣∣∣∣∣κα∑i=1

∫A(i;PN,ζ

′α )

α(x)λ(dx)−κα∑i=1

∫A(i;PN,ζ

′α )

α(x) λα(dx)

∣∣∣∣∣≤

κα∑i=1

∣∣∣∣∣∫A(i;PN,ζ

′α )

α(x)λ(dx)−∫A(i;PN,ζ

′α )

α(x) λα(dx)

∣∣∣∣∣=

κα∑i=1

∣∣∣∣∣∫A(i;PN,ζ

′α )

α(x)λ(dx)− α(t(i− 1;PN,ζ

′

α

)λ(

[t(i− 1;PN,ζ′

α ), t(i;PN,ζ′

α )))∣∣∣∣∣

=κα∑i=1

∣∣∣∣∫A(i;PN,ζ

′α )

α(x)− α(t(i− 1;PN,ζ

′

α

)λ(dx)

∣∣∣∣ ≤ κα∑i=1

∫A(i;PN,ζ

′α )

∣∣∣α(x)− α(t(i− 1;PN,ζ

′

α

)∣∣∣λ(dx)


(A.32)

≤ ε

2KN

κα∑i=1

∫A(i;PN,ζ

′α )

λ(dx) =ε

2KNλ ([0, N ]) . (A.33)

For every k ∈ N and α ∈ A we define µαk := (µk)α

. Now, using the approximation (A.33) for µk, for

k ∈ N, we obtain

supk∈N,α∈A

∣∣∣∣∫[0,N ]

α(x)µk(dx)−∫

[0,N ]

α(x) µαk (dx)

∣∣∣∣ < ε

2. (A.34)

To sum up, we have constructed the measures µαk such that the first and third summand of (A.31) become

arbitrarily small uniformly on k ∈ N and on α ∈ A. We can conclude (A.27) once we obtain the validity of

limk→∞

supα∈A

∣∣∣∣∫[0,N ]


[0,N ]

α(x) µα∞(dx)

∣∣∣∣ = 0.

Indeed, for fixed ζ ′ ∈ (0, ζ),∣∣∣∣∫[0,N ]


[0,N ]

α(x) µα∞(dx)

∣∣∣∣ =

∣∣∣∣ κα∑

i=1

∫A(i;PN,ζ

′α )

α(x) µαk (dx)−κα∑i=1

∫A(i;PN,ζ

′α )

α(x) µα∞(dx)

∣∣∣∣=

∣∣∣∣ κα∑

i=1

(∫A(i;PN,ζ

′α )

α(x) µαk (dx)−∫A(i;PN,ζ

′α )

α(x) µα∞(dx))∣∣∣∣

=

∣∣∣∣ κα∑

i=1

α(t(i;PN,ζ

′

α ))[µk(A(i;PN,ζ

′

α ))− µ(A(i;PN,ζ

′

α ))]∣∣∣∣

(ii)

≤ ‖α‖∞κα∑i=1

∣∣∣µk(A(i;PN,ζ′

α ))− µ∞

(A(i;PN,ζ

′

α ))∣∣∣

≤ supα∈A‖α‖∞ · κα · sup

i=1,...,κα

∣∣∣µk(A(i;PN,ζ′

α ))− µ∞

(A(i;PN,ζ

′

α ))∣∣∣

≤ supα∈A‖α‖∞ · κα · sup

I∈IN

∣∣µk(I)− µ∞(I)∣∣. (A.35)

Let us now denote m(PN,ζ′

α ) := mint(i;PN,ζ

′

α )− t(i− 1;PN,ζ′

α ), i = 1, . . . , κα> ζ ′ by the definition of

PN,ζ′

α . Therefore,

supα∈A

κα ≤ supα∈A

⌈ N

m(PN,ζ′

α )

⌉5 ≤

⌈Nζ ′

⌉<∞.

Hence, for every fixed sparse set PN,ζ′

α which satisfies (A.32) we obtain

supα∈A

∣∣∣∣∫[0,N ]


[0,N ]

α(x) µα∞(dx)

∣∣∣∣ ≤ supα∈A‖α‖∞ · sup

α∈Aκα · sup

I∈IN

∣∣µk (I)− µ (I)∣∣ (i)−−−−→k→∞

0,

where we have used Proposition A.12.(V) which is equivalent to (i).

Remark A.15. We have presented the previous theorem for A ⊂ D(R). However, the result can bereadily adapted for A ⊂ D(Rp).

Proposition A.16. Let the measurable space(R+,B(R+)

)and (µk)k∈N be a sequence of finite measures

such that µk(0) = 0 for every k ∈ N and µ∞ is atomless. Additionally, let A := (αk)k∈N ⊂ D(Rp).Assume the following to be true:

(i) µkw−−→ µ∞.

(ii) A is uniformly bounded, i.e. supα∈A ‖α‖∞ <∞.

(iii) A is δJ1(Rp)−convergent. In other words, αkJ1(Rp)−−−−−→k→∞

α∞.

Then,

limk→∞

∥∥∥∥∫[0,∞)

αk(x)µk(dx)−∫

[0,∞)

α∞(x)µ∞(dx)

∥∥∥∥2

= 0.

5We denote by dxe the least integer greater than or equal to x.


Proof. Let us denote

γk,m :=

∥∥∥∥∫[0,∞)

αk(x)µm(dx)−∫

[0,∞)

αk(x)µ∞(dx)

∥∥∥∥2

for k ∈ N and m ∈ N.

We are going to apply Theorem A.9 in order to obtain the required result. By Theorem A.14 we havethat

limm→∞

supk∈N

∣∣γk,m∣∣ = 0.

In other words, Condition (i) of Theorem A.9 is satisfied. Let us prove, now, that Condition (ii) of theaforementioned theorem is also satisfied, i.e. we need to prove that

limm→∞

(lim supk→∞

γk,m − lim infk→∞

γk,m)

= 0.

However, it is sufficient to prove that limm→∞ lim supk→∞ γk,m = 0, since the elements of the doubly-indexed sequence are positive. To this end, we have

lim supk→∞

γk,m

≤ lim supk→∞

∥∥∥∥∫[0,∞)

αk(x)µm(dx)−∫

[0,∞)

α∞(x)µm(dx)

∥∥∥∥2

+ lim supk→∞

∥∥∥∥∫[0,∞)

α∞(x)µm(dx)−∫

[0,∞)

α∞(x)µ∞(dx)

∥∥∥∥2

+ lim supk→∞

∥∥∥∥∫[0,∞)

α∞(x)µ∞(dx)−∫

[0,∞)

αk(x)µ∞(dx)

∥∥∥∥2

.

We are going to prove now that the two last summands of the right-hand side of the above inequality

are equal to zero. We start with the second and we realise that we have only to use that µkw−−→ µ∞ and

Theorem A.14 for the special case of a singleton, in order to verify our claim. The third summand is alsoequal to zero as an outcome of the bounded convergence theorem. Indeed, we have that the the sequenceA is uniformly bounded and by Proposition I.121 we have the pointwise convergence αk(x) −−→ α∞(x)for k → ∞, at every point x which is point of continuity of α∞. Since the set x ∈ R+,∆α

∞(x) 6= 0is at most countable, we can conclude that it is a µ∞−null set. Therefore, since every element of A isBorel measurable, we can conclude.

Finally, we deal with the first summand. Let us provide initially some helpful results. Using againProposition I.121 we have that

lim supk→∞

∥∥αk(x)− α∞(x)∥∥

2≤∥∥∆α∞(x)

∥∥2

for every x ∈ R+, (A.36)

because the only possible accumulation points of the sequence(αk,i(x)

)k∈N are α∞,i(x) and α∞,i(x−),

for every i = 1, . . . , p and every x ∈ R+; recall Remark I.119. Now observe that the function R+ 3 x 7−→∥∥∆α∞(x)∥∥

2∈ R+ is Borel measurable and µ∞−almost everywhere equal to the zero function. This

observation allows us to apply Mazzone [51, Theorem 1] in order to conclude that∫[0,∞)

∥∥∆α∞(x)∥∥

2µm(dx) −−−−→

m→∞

∫[0,∞)

∥∥∆α∞(x)∥∥

2µ∞(dx) = 0. (A.37)

In view of the above and the boundedness of the sequence A, we apply Fatou’s lemma and we obtain forevery m ∈ N

lim supk→∞

∥∥∥∥∫[0,∞)

αk(x)µm(dx)−∫

[0,∞)

α∞(x)µm(dx)

∥∥∥∥2

≤∫

[0,∞)

lim supk→∞

∥∥∥αk(x)− α∞(x)∥∥∥

2µm(dx)

(A.36)

≤∫

[0,∞)

∥∥∆α∞(x)∥∥

2µm(dx).

Now, we can conclude that the Condition (ii) of Theorem A.9 is indeed satisfied by combining the abovebound with Convergence (A.37).

Remark A.17. Let us adopt the notation of Theorem A.14 and Proposition A.16 and assume that themeasurable space is

([0, T ],B([0, T ])

), for some T ∈ R+. We claim that we can adapt the aforementioned

results without loss of generality, which can be justified as follows. The limit measure µ∞ is atomless, sothe generality is not harmed if in Theorem A.14 the integrands αk, for some k ∈ N, have a jump at point T .


Indeed, by the weak convergence µk ⇒ µ∞ and Proposition A.12 we have that the sequence of distributionfunctions (Ck)k∈N which is associated to the sequence (µk)k∈N converges uniformly. Therefore, we caneasily conclude that

lim supk→∞

∥∥∆αk(T )∥∥

2µk(T) = lim sup

k→∞

∥∥∆αk(T )∥∥

2∆CkT ≤

∥∥∆α∞(T )∥∥

2∆C∞T = 0.

In other words, we can simply assume that the distribution functions are constant after time T in orderto reduce the general case to the compact-interval case.

The following corollary is almost evident due to the fact that the limit measure is atomless. Howeverwe state it in the form we will need it in Subsection IV.5.2 and we provide its complete (rather trivial)proof.

Corollary A.18. Let the measurable space(R+,B(R+)

), (µk)k∈N be a sequence of finite measures, where

µk(0) = 0 for every k ∈ N and µ∞ is atomless. Let, moreover, A := (αk)k∈N ⊂ D(Rp) and the followingto be true:

(i) µkw−−→ µ∞.

(ii) A is uniformly bounded, i.e. supα∈A ‖α‖∞ <∞ and(iii) A is δJ1(Rp)−convergent.

Then, ∫[0,·]

αk(x)µk(dx)J1(Rp)−−−−−→k→∞

∫[0,·]

α∞(x)µ∞(dx). (A.38)

Proof. Let Ck denote the distribution function associated to the measure µk for every k ∈ N and

γk(t) :=

∫[0,t]

αk(x)µk(dx) =

∫[0,t]

αk(s) dCks ∈ D(Rp) for k ∈ N.

We are going to apply Proposition I.118 in order to prove the required convergence. To this end, letus fix a t∞ ∈ R+ and a sequence (tk)k∈N such that tk −−→ t∞ as k → ∞. We need to prove that therequirements (i) - (iii) of the aforementioned theorem are satisfied.

(i) limk→∞

(∥∥γk(tk)−γ∞(t∞)∥∥

2∧∥∥γk(tk)−γ∞(t∞−)

∥∥2

)= 0. However, γ∞ is continuous. Therefore,

it is sufficient to prove

limk→∞

∥∥γk(tk)− γ∞(t∞)∥∥

2= 0.

To this end, we have

γk(tk) =

∫[0,tk]

αk(s) dCks =

∫[0,tk∧t∞]

αk(s) dCks +

∫[tk∧t∞,t∞]

αk(s) dCks +

∫[t∞,tk∧t∞]

αk(s) dCks .

The first summand converges to γ∞(t∞), because the sequence (Ck1[0,tk])k∈N is ‖ · ‖∞−convergent and

(αk)k∈N is uniformly bounded δJ1(Rp)−convergent. In other words, the conditions of Proposition A.16are satisfied. For the second summand we have∣∣∣∣∫

[tk∧t∞,t∞]

αk(s) dCks

∣∣∣∣ ≤ Var

(∫[0,·]

αk(s) dCks

)t∞−Var

(∫[0,·]

αk(s) dCks

)tk∧t∞

=

∫[tk∧t∞,t∞]

(αk(s) ∨ 0

)dCks +

∫[tk∧t∞,t∞]

(− αk(s) ∨ 0

)dCks

≤ supk∈N

∥∥αk∥∥∞µk([tk ∧ t∞, t∞])

= supk∈N

∥∥αk∥∥∞(Ckt∞ − Cktk∧t∞) −−−−→k→∞0,

where we used Proposition A.12 in order to conclude the convergence. Analogously we can prove thatthe third summand converges also to 0.

(ii) If limk→∞∥∥γk(tk)−γ∞(t∞)

∥∥2

= 0 and (sk)k∈N such that tk ≥ sk for every k ∈ N and sk −−−−→k→∞

t∞,

then

limk→∞

∥∥γk(sk)− γ∞(t∞)∥∥

2= 0.

We can repeat the exact same arguments as in (i) in order to prove the convergence for every (sk)k∈Nsuch that sk −−−−→

k→∞t∞.


(iii) If limk→∞∥∥γk(tk)− γ∞(t∞−)

∥∥2

= 0 and (sk)k∈N such that sk ≥ tk for every k ∈ N and sk −−−−→k→∞

t∞, thenlimk→∞

∥∥γk(sk)− γ∞(t∞−)∥∥

2= 0.

due to the continuity of γ∞, we can apply the same argument as in the previous cases.

The following lemma will be helpful in proving that a relatively compact set of D(Rp) is uniformlybounded. Observe that every relatively compact subset of D(Rp) satisfies (ii) of the following lemma; seeTheorem I.111.

Lemma A.19. Let (αk)k∈N ⊂ D(Rp)

such that

(i) αk(∞) := limt→∞ αk(t) exists and∥∥αk(∞)‖2 <∞,

(ii) supk∈N supt∈[0,N ]

∥∥αk(t)∥∥

2<∞ for every N ∈ N, and

(iii) lim sup(k,t)→(∞,∞)

∥∥αk(t)∥∥

2<∞.

Then the sequence (αk)k∈N is uniformly bounded, i.e. supk∈N ‖α‖∞ <∞.

Proof. Assume that the sequence is unbounded, i.e. for every M ∈ N there exists k(M) ∈ N suchthat ‖αk(M)‖∞ > M . We can extract, then, a subsequence of

(k(M)

)M∈N, name

(k(Mn)

)n∈N, such that

k(Ml) < k(Mm) whenever l < m and Mn → ∞ as n → ∞. This can be done as follows. For n = 1 wedefine k(M1) := k(1) and for i ∈ N \ 1 we set

k(Mi+1) := mink(M) ∈ N, ‖αk(M)‖∞ >

⌈‖αk(Mi)‖∞

⌉,

where dxe denotes the least integer greater than or equal to x. Since we have assumed that for everyk ∈ N the value

∥∥αk(∞)‖2 is finite, we have in particular (using that αk ∈ D(Rp)) that ‖αk‖∞ < ∞.Therefore,

k(M) ∈ N, ‖αk(M)‖∞ >⌈‖αk(Mi)‖∞

⌉6= ∅

and its minimum exists as N is a well-ordered set under the usual order. In view of the these comments, thesubsequence

(k(Mn)

)n∈N is well-defined and, according to our initial assumption, it has to be unbounded.

By definition of αk(M), there exists tk(M) ∈ R+ such that ‖αk(M)(tk(M))‖∞ > M , a property whichholds in particular for the subsequence indexed by (k(Mn))n∈N. Let us distinguish, now, two cases forthe sequence (tk(Mn))n∈N.

If (tk(Mn))n∈N is bounded, then (ii) leads to a contradiction, for N := suptk(Mn), n ∈ N. If (tk(Mn))n∈N is unbounded, then there exists a subsequence (tk(Mnl

))l∈N such that tk(Mnl) −−→∞

as l→∞. Therefore, we have also

liml→∞

∥∥αk(Mnl)(tk(Mnl

))∥∥∞ ≥ lim

l→∞Mnl =∞.

But, then (iii) leads to a contradiction.Now, we have only to verify that the set ‖αk‖∞, k = 1, . . . , k(1) is bounded in order to conclude

that the sequence (αk)k∈N is ‖·‖∞-bounded. But the above is clear since it is a finite set of finite numbers;use that ‖αk(∞)‖2 <∞ and that αk ∈ D(Rp).

Bibliography

[1] S. Aazizi. Discrete-time approximation of decoupled forward-backward stochastic differential equa-tions driven by pure jump levy processes. Adv. in Appl. Probab., 45(3):791–821, 09 2013.

[2] D. Aldous. Stopping times and tightness. Ann. Probab., 6(2):335–340, 04 1978.[3] D. Aldous. Weak convergence and the general theory of processes. Incomplete draft, University of

California, Berkeley, 1981.[4] C. D. Aliprantis and K. Border. Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer, 3rd

edition, 2007.[5] V. Bally. Approximation scheme for solutions of bsde. In N. El Karoui and L. Mazliak, editors,

Backward stochastic differential equations, volume 364 of Chapman & Hall/CRC Research Notes inMathematics Series, pages 177 – 191. Longman, 1997.

[6] E. Bandini. Existence and uniqueness for backward stochastic differential equations driven by arandom measure, possibly non quasi-left continuous. Electronic Communications in Probability, 20,2015.

[7] P. Billingsley. Convergence of probability measures. Wiley Series in Probability and Statistics:Probability and Statistics. John Wiley & Sons Inc., New York, second edition, 1999. A Wiley-Interscience Publication.

[8] J.-M. Bismut. Analyse Convexe et Probabilites. PhD thesis, Faculte des Sciences de Paris, 1973.[9] J.-M. Bismut. Conjugate convex functions in optimal stochastic control. Journal of Mathematical

Analysis and Applications, 44(2):384–404, 1973.[10] V. Bogachev. Measure Theory. Analysis. Springer-Verlag, Berlin Heidelberg, 2007.[11] B. Bouchard and R. Elie. Discrete-time approximation of decoupled Forward–Backward SDE with

jumps. 118:53 – 75, 01 2008.[12] B. Bouchard, R. Elie, and N. Touzi. Discrete-time approximation of BSDEs and probabilistic schemes

for fully non-linear pdes. In H. Albrecher, W. Runggaldier, and W. Schachermayer, editors, AdvancedFinancial Modelling, Radon Series on Computational and Applied Mathematics, pages 91 – 124. DeGruyter, 2009.

[13] B. Bouchard, D. Possamaı, X. Tan, and C. Zhou. A unified approach to a priori estimates forsupersolutions of BSDEs in general filtrations. Annales de l’Institut Henri Poincare, Probabilites etStatistiques (B), to appear, 2015.

[14] P. Briand, B. Delyon, and J. Memin. Donsker-type theorem for BSDEs. Electron. Commun. Probab.,6:1–14, 2001. doi: 10.1214/ECP.v6-1030. URL http://dx.doi.org/10.1214/ECP.v6-1030.

[15] P. Briand, B. Delyon, and J. Memin. On the robustness of backward stochastic differential equations.Stochastic Processes and their Applications, 97:229–253, 2002.

[16] R. Buckdahn. Backward stochastic differential equations driven by a martingale. preprint, 1993.[17] R. Carbone, B. Ferrario, and M. Santacroce. Backward stochastic differential equations driven by

cadlag martingales. Theory of Probability & its Applications, 52(2):304–314, 2008.[18] D. Chevance. Resolution numeriques des equations differentielles stochastiques retrogrades,. PhD

thesis, Universite de Provence – Aix – Marseille I, 1997.[19] S. Cohen and R. Elliott. Existence, uniqueness and comparisons for BSDEs in general spaces. The

Annals of Probability, 40(5):2264–2297, 2012.[20] S. Cohen and R. Elliott. Stochastic Calculus and Applications. Springer, 2nd edition, 2015.[21] S. Cohen, R. Elliott, and C. Pearce. A general comparison theorem for backward stochastic differ-

ential equations. Advances in Applied Probability, 42(3):878–898, 2010.[22] F. Confortola, M. Fuhrman, and J. Jacod. Backward stochastic differential equation driven by a

marked point process: An elementary approach with an application to optimal control. Ann. Appl.Probab., 26(3):1743–1773, 06 2016.

149

http://dx.doi.org/10.1214/ECP.v6-1030

150 BIBLIOGRAPHY

[23] F. Coquet, J. Memin, and L. S lominski. On weak convergence of filtrations. Seminaire de Probabilites,XXXV:306–328, 2001.

[24] M. Davis and P. Varaiya. Dynamic programming conditions for partially observable stochasticsystems. SIAM Journal on Control and Optimization, 11(2):226–261, 1973.

[25] M. Davis and P. Varaiya. The multiplicity of an increasing family of σ−fields. The Annals ofProbability, 2(5):958–963, 1974.

[26] C. Dellacherie and P.-A. Meyer. Probabilities and Potential. North-Holland, 1978.[27] C. Dellacherie and P.-A. Meyer. Probabilities and Potential, B: Theory of Martingales. North-

Holland, 1982.[28] R. Dudley. Real Analysis and Probability. Cambridge University Press, 2nd edition, 2002.[29] N. El Karoui and S.-J. Huang. A general result of existence and uniqueness of backward stochastic

differential equations. In N. El Karoui and L. Mazliak, editors, Backward stochastic differentialequations, volume 364 of Chapman & Hall/CRC Research Notes in Mathematics Series, pages 27–36. Longman, 1997.

[30] N. El Karoui and L. Mazliak. Backward Stochastic Differential Equations. Chapman & Hall/CRCResearch Notes in Mathematics Series. Taylor & Francis, 1997.

[31] N. El Karoui, S. Peng, and M.-C. Quenez. Backward stochastic differential equations in finance.Mathematical Finance, 7(1):1–71, 1997.

[32] P. Embrechts and M. Hofert. A note on generalized inverses. Mathematical Methods of OperationsResearch, 77(3):423–432, 2013.

[33] S. Ethier and T. Kurtz. Markov Processes: Characterization and Convergence. J. Wiley & Sons,1986.

[34] H. Follmer and P. Protter. Local martingales and filtration shrinkage. ESAIM Probab. Stat., 15(Inhonor of Marc Yor, suppl.):S25–S38, 2011. ISSN 1292-8100.

[35] S. He, J. Wang, and J. A. Yan. Semimartingale Theory and Stochastic Calculus. CRC Press, 1992.[36] P. Henry-Labordere, X. Tan, and N. Touzi. A numerical algorithm for a class of BSDEs via the

branching process. Stochastic Processes and their Applications, 124(2):1112 – 1140, 2014. ISSN0304-4149.

[37] E. Hobson. The theory of functions of a real variable and the theory of Fourier’s series. Cambridge:University Press, 1907.

[38] D. Hoover. Convergence in distribution and Skorokhod convergence for the general theory of pro-cesses. Probability Theory and Related Fields, 89(3):239–259, 1991.

[39] Y. Hu and S. Peng. A stability theorem of backward stochastic differential equations and its ap-plication. Comptes Rendus de l’Academie des Sciences, Serie I: Mathematique, 324(9):1059–1064,1997.

[40] J. Jacod. Calcul Stochastique et Problemes de Martingales. Lecture Notes in Mathematics. SpringerBerlin Heidelberg, 2006.

[41] J. Jacod and A. Shiryaev. Limit Theorems for Stochastic Processes. Springer, 2nd edition, 2013.[42] J. Jacod, S. Meleard, and P. Protter. Explicit form and robustness of martingale representations.

The Annals of Probability, 28:1747–1780, 2000.[43] M. Krasnosel’skii and Y. Rutickii. Convex Functions and Orlicz Spaces. P. Noordfoff Ltd., Groningen,

1961.[44] T. Kurtz and P. Protter. Weak limit theorems for stochastic integrals and stochastic differential

equations. The Annals of Probability, 19(3):1035–1070, 07 1991.[45] R. Leadbetter, S. Cambanis, and V. Pipiras. A Basic Course in Measure and Probability: Theory

for Applications. Cambridge University Press, 2014.[46] A. Lejay and M. S. Torres. Numerical approximation of backward stochastic differential equations

with jumps, 2013.[47] E. Lenglart, D. Lepingle, and M. Pratelli. Presentation unifiee de certaines inegalites de la theorie

des martingales. Seminaire de Probabilites, XIV:26–52, 1980.[48] R. Long. Martingale Spaces and Inequalities. Vieweg+Teubner Verlag, 1993.[49] J. Ma, P. Protter, J. San Martin, and S. Torres. Numberical method for backward stochastic

differential equations. The Annals of Applied Probability, 12(1):302–316, 2002.[50] D. Madan, M. Pistorius, and M. Stadje. Convergence of BS∆Es driven by random walks to BSDEs:

the case of (in)finite activity jumps with general driver. Stochastic Processes and their Applications,126:1553–1584, 2016.

BIBLIOGRAPHY 151

[51] F. Mazzone. A characterization of almost everywhere continuous functions. Real Analysis Exchange,21(1):317–319, 1995.

[52] P.-A. Meyer. Sur le lemme de la Vallee Poussin et un theoreme de Bismut. Seminaire de Probabilites,XII:770–774, 1978.

[53] J. Memin. Stability of Doob–Meyer decomposition under extended convergence. Acta MathematicaeApplicatae Sinica, 19:177–190, 2003.

[54] A. Papapantoleon, D. Possamaı, and A. Saplaouras. Existence and uniqueness results for BSDEswith jumps: the whole nine yards. Preprint, arXiv:1607.04214, 2016.

[55] E. Pardoux and S. Peng. Adapted solution of a backward stochastic differential equation. Systemand Control Letters, 14(1):55–61, 1990.

[56] K. Parthasarathy. Probability Measures on Metric Spaces. Academic Press, 1972.[57] Y. V. Prokhorov. Convergence of random processes and limit theorems in probability theory. Theory

of Probability and Applications, I(2):157 – 214, 1956.[58] M. Rao and Z. Ren. Theory of Orlicz Spaces. Marcel Dekker, 1991.[59] R. Rebolledo. La methode des martingales appliquee a l’etude de la convergence en loi de processus.

Memoires de la Societe Mathematique de France, 62:I1–V125, 1979.[60] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Grundlehren der mathema-

tischen Wissenschaften. Springer Berlin Heidelberg, 2004.[61] R. Rockafellar. Convex Analysis. Princeton landmarks in mathematics and physics. Princeton

University Press, 1970.[62] M. Royer. BSDEs with a random terminal time driven by a monotone generator and their links with

PDEs. Stochastics and Stochastic Reports, 76(4):281–307, 2004.[63] W. Rudin. Principles of Mathematical Analysis. WCB/McGraw-Hill, third edition, 1976.[64] K. Sato. Levy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced

Mathematics. Cambridge University Press, 1999. ISBN 9780521553025.[65] J. Zhang. A numerical scheme for bsdes. Ann. Appl. Probab., 14(1):459–488, 02 2004.

backward stochastic differential equations with jumps are ... › bitstream › 11303 › ... ·...

Documents