reflected backward stochastic differential …€¦ · sator is not absolutely continuous, but...

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POLITECNICO DI MILANODEPARTMENT OF MATHEMATICS

DOCTORAL PROGRAMME IN MATHEMATICALMODELS AND METHODS FOR ENGINEERING

REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL

EQUATIONS DRIVEN BY MARKED POINT PROCESSES AND

APPLICATIONS TO OPTIMAL STOPPING AND OPTIMAL

SWITCHING

Doctoral Dissertation of:Nahuel Foresta

Advisor:Prof. Marco Fuhrman

The Chair of the Doctoral Program:Prof. Irene Sabadini

2017 – XXX Cycle

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Abstract

In the present work, we focus on some control problems for a rather general class ofmarked point processes. We assume these processes to be non-explosive and quasi-leftcontinuous, which translate to having a continuous compensator. In particular we de-fine and treat optimal stopping and optimal switching problems where the randomnesscomes from this type of processes, as well from a diffusive part. The problems areformulated in a general non-markovian way, so a natural tool to solve them are back-ward stochastic differential equations. Thus we establish, under suitable assumptions,existence and uniqueness results for a class of reflected BSDE driven by an integerrandom measure and a Wiener process. This is done first in the case of a given cadlagobstacle, and later in the case where these equations form a system and the obstacledepends on the solutions to the other components to the system. Due to the nature ofthe compensator of the point process, we solve the problem in “weighted” L2 spaces,where the weight depends on the compensator. The solutions to these equations are thenused to represent the value function to an optimal stopping problem and to an optimalswitching problem, respectively. In the latter problem, we allow the switching mode tomodify the dynamic of the point process through a change in its compensator and thusin the probability under which we consider it. In both cases we obtain the usual char-acterizations of the optimal strategies as hitting times of the solution into the obstacle.In the optimal stopping problem, we also consider the control randomization approachand related constrained BSDE. In the process of obtaining these results we also extendsome other typical tools in BSDE theory to the case of marked point processes, like thecomparison theorem or a simple version of the monotonic limit theorem.

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Preface

The work presented in this thesis has been developed during my three years as a Ph.Dstudent in Politecnico di Milano.

The results included in chapter 3 are taken from Foresta [39], which has been sub-mitted to a journal. The work has also been presented in a talk at the “First ItalianMeeting in Probability and Statistics”, which was held in June 2017 at Turin.

The results in chapter 6, along with other support results present in the thesis, arecontained in Foresta [40]. This work was presented in a short talk at the “Workshopin BSDE and SPDE” of July 2017 at Edinburgh, and in a poster at the “Conference onStochastic Control, Ambiguity and Games” held in Leeds on September 2017.

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Acknowledgements

I would like to express my sincere gratitude to my advisor Prof. Marco Fuhrman foroffering me great guidance and advice in the creation of this Ph.D. thesis. His recom-mendations have been precious in steering me towards interesting mathematical prob-lems and in teaching me how to tackle them. I would also like to thank Dr. FulviaConfortola for the valuable discussions throughout the years.

I am also very grateful to Prof. Ying Hu and Prof. Idris Kharroubi, for agreeing tobe my referees and for their useful comments. I also would like to thank Prof. HelmuthAbels, Prof. Idris Kharroubi and Prof. Gianmaria Verzini for taking part in my thesisdefense committee.

Finally, I gratefully acknowledge all the people who accompanied me through thisjourney and made the completion of this thesis possible.

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Contents

Abstract I

Preface III

Aknowledgements V

1 Introduction 11.1 Summary of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Introduction to the mathematical setting and complements on BSDEdriven by marked point processes . . . . . . . . . . . . . . . . . 2

1.1.2 Reflected BSDE . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 Randomization and constrained BSDE . . . . . . . . . . . . . . 71.1.4 A switching problem for marked point processes . . . . . . . . . 9

2 Mathematical tools 132.1 Marked point processes . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Integrals with respect to marked point processes . . . . . . . . . 142.1.2 Changes of probability . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Probabilistic setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Backward stochastic differential equations . . . . . . . . . . . . . . . . 22

2.3.1 A monotonic limit result . . . . . . . . . . . . . . . . . . . . . . 302.4 Optimal stopping and optimal switching . . . . . . . . . . . . . . . . . 332.5 Some remarks on the Snell envelope theory . . . . . . . . . . . . . . . 35

3 RBSDE driven by MPP and optimal stopping 393.1 Preliminaries, assumptions, formulation of the problems . . . . . . . . 413.2 Reflected BSDE with known generators and optimal stopping problem . 433.3 Reflected BSDE in the general case . . . . . . . . . . . . . . . . . . . 55

4 Additional results on the reflected backward SDE 594.1 Approximating the RBSDE . . . . . . . . . . . . . . . . . . . . . . . . 604.2 A comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 65

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Contents

5 Randomization technique for Optimal stopping and constrained BSDE 675.1 The constrained BSDE . . . . . . . . . . . . . . . . . . . . . . . . . . 685.2 The dual problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6 A switching problem for marked point processes 776.1 Preliminaries and formulation of the problem . . . . . . . . . . . . . . 786.2 Existence of a solution to the system of RBSDE . . . . . . . . . . . . . 816.3 Verification theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7 Final remarks and future work 101

Bibliography 103

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CHAPTER1Introduction

In this preliminary chapter we give a general outline of the dissertation. After a shortintroduction to the mathematical topics relevant to this work, we briefly describe ourmain results and the structure of the subsequent chapters. In this introduction we justgive an a summary of the main results and the main hypotheses, leaving the detaileddescription of the literature, setting and technicalities to the relevant chapters.

In the present PhD thesis we deal with optimal control of stochastic systems. Thefirst ingredient in this field is some random dynamics, which might depend on someparameter henceforth called control variable. The second is some functional of the stateof the system that might also depend on the control variable. An agent is interestedin maximizing or minimizing the expected value of the functional over all possibleoutcomes of the system evolutions and over all possible choices of the control variableover time. General references to this field can be found in Krylov [62], Bensoussan [8],Fleming and Soner [38] or Pham [72] among others.

In this work we concentrate on a particular type of jump processes in continuoustime, namely marked point processes. Basically a point process is a sequence of pairs(Tn, ξn)n≥0, where Tn is a sequence of strictly increasing jump times and ξn is a randommark that takes value in some measurable space E. Such a process defines an integerrandom measure p(dsde) that counts the number of pairs in subsets of R+ × E. Acentral notion in this theory is the one of compensator, a predictable measure ν(dsde)on R+ × E, such that integrals of predictable processes with respect to p− ν are mar-tingales (or local martingales). In particular it can be written as φs(de)dAs, where φis a transition kernel from Ω × R+ into E and A is the dual predictable projection ofthe counting process N associated to p. In particular, we will write down backwardstochastic differential equations (backward SDE or BSDE) were the driving noise is an

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Chapter 1. Introduction

integer random measure associated to the marked point process.The subject of optimal control of marked point process has been studied in the past,

and general references can be found for example in Bremaud [10]. On the other hand,the use of backward SDE technique to approach these problems is relatively new. InCohen and Elliott [16] we have a detailed exposition of weak control of point processesin the case of single jump and in the case of compensator absolutely continuous withrespect to the Lebesgue measure in time, the latter also treated with BSDE techniques.The work Confortola and Fuhrman [19] tackles a similar problem when the compen-sator is not absolutely continuous, but still it is required to be continuous in time. Tothis end they define a class of BSDE driven by a marked point process martingale, thegenerator being integrated with respect to the process A. The setting of this thesis isvery similar, but we also consider the presence of a Wiener diffusion part.

Nonlinear backward stochastic differential equations (BSDE) driven by a Brownianmotion were first introduced by Pardoux and Peng in the seminal paper Pardoux andPeng [66]. Later, BSDE have found applications in several fields of mathematics, suchas stochastic control, mathematical finance, nonlinear PDEs (see for instance El Karoui,Peng, and Quenez [29], Pardoux and Rascanu [68], Crepey [21], Pardoux [67]). Asthe driving noise, the Brownian motion has been replaced by more general classes ofmartingales; the first example is perhaps El Karoui and Huang [28], see Xia [80] for avery general situation.

In particular, occurrence of jumps in the equation has been considered since long.In Tang and Li [78] and Barles, Buckdahn, and Pardoux [4], related to optimal controland PDEs respectively, an independent Poisson random measure is added to the drivingWiener noise. Motivated by several applications to stochastic optimal control and fi-nancial modelling, more general marked point processes were considered in the BSDE.Examples can be found in Becherer [5], Confortola and Fuhrman [19] for L2 solutions,Confortola, Fuhrman, and Jacod [20] for the L1 case and Confortola [18] for the Lp

case.

1.1 Summary of the thesis

Now we give an overview of the problems treated in this thesis and the main results weachieve. All the proofs in the present work are original, while the results by authors arestated without proof. The aim of this section is to illustrate the main hypotheses andresults, as well as the methods we have used to obtain them. A more detailed analysisof the literature can be found in the respective chapters, while here we only cite themost closely related works.

1.1.1 Introduction to the mathematical setting and complements on BSDE drivenby marked point processes

Before delving into the results of the thesis, it is worth to spend a few lines to introducethe mathematical tools we deal with in order to make the following subsections morereadable. The precise version of this introduction constitutes chapter 2.

On (Ω,F ,P) we are given a non-explosive marked point process p whose markstake value in some measurable space (E, E) and an independent Brownian motion

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1.1. Summary of the thesis

W . We work with the completed filtration generated by p and W . As mentionedbefore the associated random measure p(ω, dsde) admits a compensator of the formφs(ω, de)dAs(ω). Together they define a compensated measure q(ω, dsde) = p(ω, dsde)−φs(ω, de)dAs(ω), that has the property that for every C ∈ E , q((0, t]×C) is a martin-gale or local martingale. By integrating processes that are P ⊗E-measurable functions(P denotes the predictable σ-algebra) with respect to q we obtain a class of purelydiscontinuous martingales. We make the following assumption on the process A

Assumption : The process A is continuous.

More details about the marked point processes can be found in section 2.1, but werecall here two important properties of our setting. The first key property for the dis-cussion that follows is the martingale representation theorem 2.10. Given a square inte-grable martingale M , there exist two processes U and Z with appropriate integrabilityproperties such that

Mt = M0 +

∫ t

0

∫E

Us(e)q(dsde) +

∫ t

0

ZsdWs.

The other important notion is a theorem of Girsanov type (see section 2.1.2), whichtells us that given a non-negative P ⊗E-measurable function ρ we can introduce a newprobability Pρ P such that the compensator of the point process under Pρ becomesρs(e)φs(de)dAs, while W is still an independent Brownian motion. In order for this tohappen we have to assume some technical conditions, for instance that ρ is bounded bysome constant L and that E

[eηAT

]<∞ for some η > 3 + L4.

With this setting in mind we introduce a BSDE of the form

Yt = ξ +

∫ T

t

fs(Ys, Us)dAs +

∫ T

t

gs(Ys, Zs)ds−∫ T

t

∫E

Us(e)q(dsde)−∫ T

t

ZsdWs

(1.1)where the drivers are both the compensated random measure and the Brownian motion.One peculiarity is the presence of two generators, where the one relative to Y and Uis integrated with respect to the measure dA. A solution to this equation is a triple(Y, U, Z) with Y cadlag that lies in a kind of L2 weighted spaces. For a more precisedefinition see section 2.2, but the idea is that we define the spaces L2,β(A), L2,β(p) andL2,β(W ) as the spaces of functions such that the respective norms are finite:

||Y ||L2,β(A) = E[∫ T

0

eβAs|Ys|2dAs]

||U ||L2,β(p) = E[∫ T

0

∫E

eβAs|Us(e)|2φs(de)dAs]

(1.2)

||Z||L2,β(W ) = E[∫ T

0

eβAs|Zs|2ds].

The solution (Y, U, Z) lies in (L2,β(A) ∩ L2,β(W )) × L2,β(p) × L2,β(W ). These L2,β

spaces depend on the parameter β, whose role will be clear in the following. A similarequation was studied in Confortola and Fuhrman [19], but in the case where there is no

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Brownian motion. In this work we prove the result when also a diffusive part is added.To establish wellposedness, we require that

• f and g are Lipschitz in (Y, U) and (Y, Z) respectively.

• E[eβAT |ξ|2

]<∞, ft(0, 0) ∈ L2,β(A) and gt(0, 0) ∈ L2,β(W ).

The precise hypotheses (2-N) and (2-N’) under which this equation has a solution canbe found in section 2.3. In particular we need the parameter β of the spaces in whichthe data lie to be larger than a threshold that depends on the Lipschitz constants off . The proof to establish existence and uniqueness is classical in BSDE theory: whenf and g do not depend on the solution, we define a martingale from the data and weuse the martingale representation theorem to find Y , U and Z. Then we have to provethat these processes are indeed in the spaces defined above by applying the Ito formulato eβAt |Yt|2 and using some direct estimates. Note that since we have integrals withrespect to dA some standard arguments can not be applied. The general case is obtainedthrough the use of the contraction theorem.

After obtaining wellposedness of such equation, we prove a comparison theorem(theorem 2.15). It is well known (see Barles, Buckdahn, and Pardoux [4]) that a generalcomparison theorem does not hold in the presence of jumps. We thus have to addan additional requirement on the generator f and the process A. We make the usualassumption that, for u2, u1 ∈ L2(E, E , φt(ω, de)) it holds

f 2t (y, u2)− f 2

t (y, u1) ≤∫E

γt(e)(u2(e)− u1(e))φt(de), (1.3)

for some P⊗E-measurable function γ such that−1 ≤ γ ≤ C. We want to use changesof probability “a la Grirsanov” as described before, so we also have to assume thatE[eηAT

]<∞ for some η > 3 + (C + 1)4.

If these conditions hold, along with the conditions for the solutions to exist and beunique, then it is possible to obtain the comparison result. This means that if ξ2 ≤ ξ1,f 2t (y, u) ≤ f 1

t (y, u), g2t (y, z) ≤ g1

t (y, z) then the corresponding solutions Y 2 and Y 1

are such that Y 2t ≤ Y 1

t for all t a.s. Notice that in (1.3) we allow γ to attain the value−1. This is important for the kind of equations we want to compare later in this work,but introduces some technical complications. Since in this case the probability measurechange induced by γ is not necessarily equivalent to the reference probability it makesthe comparison argument a bit more involved, and we resort to approximations to makeit work. Details can be found in theorem 2.15. Lastly we state and prove a simplemonotonic limit theorem for BSDEs when the generator f depends only on U and in alinear manner, and g does not depend on (Y, Z).

The rest of chapter 2 gives some general notions about optimal stopping and switch-ing, as well as introducing the key notions of the Snell envelope that we will use insubsequent chapters.

1.1.2 Reflected BSDE

Starting from the work El Karoui et al. [30], reflected BSDE have been widely used tosolve optimal stopping problems and obstacle problems. In chapter 3 we introduce thefollowing reflected equation

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Yt = ξ +

∫ Ttf(s, Ys, Us)dAs +

∫ Ttg(s, Ys, Zs)ds−

∫ Tt

∫EUs(y)q(dsdy)

−∫ TtZs(y)dWs +KT −Kt, ∀t ∈ [0, T ] a.s.

Yt ≥ ht, ∀t ∈ [0, T ] a.s.∫ T0

(Ys − hs)dKcs = 0 and ∆Kt ≤ (ht− − Yt)+

1Yt−=ht−∀t ∈ [0, T ] a.s.,

(1.4)

where the unknowns are Y , U , Z and K. The driving noises in this equation are boththe martingale measure q(dsde) = p(dsde) − φs(de)dAs and the Brownian motionW , which are taken to be independent. As for the non reflected BSDE, we have thepresence of two generators, one for each noise, both satisfying an appropriate Lipschitzcondition. The Y part of the solution is constrained to stay above a given cadlag processh, and the K part is there to make this possible. The last line is called the minimal pushcondition, and tells us that K only acts when Y is about to touch the obstacle h. Asalready said, we make the assumptions that the point process is non-explosive and hasonly totally inaccessible jumps, which in turn means that the process A is continuous.The closest work to this is the paper Hamadene and Ouknine [47] in a Brownian andPoisson framework with a general cadlag barrier.

The appropriate spaces for the solution to this equation turns out to be a kind of L2

space with a weight, which are the L2,β spaces introduced in the previous section in(1.1.1) and whose precise definition can be found in section 2.2. Again we ask the data(ξ, f(·, 0, 0), g(·, 0, 0)) to be in the same spaces, for a value of the parameter β that isstrictly larger than a threshold value dependent on the Lipschitz constants of f . Theprecise hypotheses can be found in assumption (3-B). As for the barrier h, which weassume cadlag, we need a little more integrability: we assume there is a δ > 0 such that

E

[supt∈[0,T ]

e(β+δ)At |ht|2]<∞.

This kind of BSDE is related to optimal stopping problems, where the generators rep-resent running gains, the terminal condition the reward in case the controller does notstop and the barrier the early stopping reward. Thus we introduce a non Markovianoptimal stopping problem with this data. The target quantity to maximize is

J(τ ; t) = E[∫ τ∧T

t

fsdAs +

∫ τ∧T

t

gsds+ hτ1τ<T + ξ1τ≥T

∣∣∣∣ Ft]and thus the value function

v(t) = ess supτ≥t

J(τ ; t).

As expected, the solution to the reflected BSDE represents the value function v.There are two main approaches in showing existence and uniqueness of the solution

to this kind of equations. One is the use of a penalization method, where the solution isobtained as limit of a sequence of non reflected BSDE with a penalization term in thegenerator. Through the use of a comparison theorem and a monotonic limit theoremit is possible then to obtain the existence of a solution. In frameworks with jumps,the need for a comparison theorem limits the generality of the generator, so we have

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decided not to follow this path. The other classical approach is the one often referredas Picard iteration which we briefly describe here.

We start in the case where f and g do not depend on the solution, defining Y assolution to the associate optimal stopping problem. Thanks to this definition, Y +∫ t

0fsdAs +

∫ t0gsds is a supermartingale and a Snell envelope. We can then decompose

it as the difference of a martingale, which gives us the processes U and Z, and anincreasing process K. Using the known properties of the Snell envelope, it is possibleto obtain the minimal push conditions. The following step is to show that (Y, U, Z) areindeed in the L2,β spaces. This requires a bit of work, since we have to estimate theterm

E[∫ t

0

eβAshs−dKs

],

and the presence of the weight eβAt does not allow to do it in the classical ways. Similarcomputations ensure uniqueness of the solution. All the details about this are containedin proposition 3.6.

With this result in hand we can tackle the representation of the optimal stoppingproblem. The argument to prove that Yt = v(t) is quite classical, but without furtherassumptions the existence of an optimal stopping rule is not guaranteed. We establishexistence of an optimal rule when the obstacle is left upper semi continuous in expecta-tion along stopping times (left USCE), a notion introduced in Kobylanski and Quenez[60]. For the precise statements see proposition 3.12. In this case we also have that thepushing process K is continuous, and Y also has the left USCE property. If we removethe Brownian motion from the framework, then there is really no need for square inte-grability of the data in this simple case with known generators. Indeed it is possible toobtain existence and uniqueness in a sort of L1 setting, and solve an optimal stoppingproblem with L1 data through the use of the reflected BSDE, as written in propositions3.15 and 3.16.

The general case where f and g do depend on the solution is then obtained througha fixed point argument in theorem 3.17. This is the reason for which we ask β to belarger than a threshold value. Again we obtain a result for the case with both Brownianmotion and marked point process, and one for the case with only the point process (seetheorem 3.18), in both cases we using an L2 setting. BSDE driven by a marked pointprocess in a L1 setting when the generator depends on the solution has been studied inConfortola, Fuhrman, and Jacod [20], and requires a different approach.

In the subsequent chapter 4, we state and prove some preliminary technical resultsabout the reflected BSDE. We restrict ourselves to generators of a particular form, andestablish a pair of properties that will be of use in the rest of the work. We introducethe reflected BSDE linear in U

Yt = ξ +∫ TtfsdAs +

∫ TtUs(e)(ρs(e)− 1)φs(de)dAs +

∫ Ttgsds

−∫ TtUs(e)q(dsde)−

∫ TtZsdWs +KT −Kt

Yt ≥ ht∫ T0

(Yt− − ht−)dKt = 0.

(1.5)

where ξ, f, g, h are as before, and ρ is a non-negative P ⊗ E-measurable functionbounded by some constant L. The particular form of the reflected BSDE (1.5), which

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might seem unusual now, will be justified later when solving switching problems. Theaim of this part of the work is to approximate the solution to this equation with a se-quence of solutions to non reflected BSDEs with a penalization term. This is useful byitself in the randomization approach used in chapter 5, but also to establish a compari-son theorem for the reflected BSDE (1.5).

The idea is to define the following sequence of standard BSDE (as the ones describedin section 1.1.1)

Y nt = ξ +

∫ T

t

fsdAs +

∫ T

t

∫E

Uns (e)(ρs(e)− 1)φs(de)dAs +

∫ T

t

gsds

−∫ T

t

Uns (e)q(dsde)−

∫ T

t

Zns dWs +Kn

T −Knt , (1.6)

where Knt = n

∫ t0

(Y ns − hs)

− ds. This admits a solution in L2,β spaces, if the datais integrable enough, where enough means larger than a constant depending on C, thebound for ρ. We show that Y n ≤ Y n+1, and Y n Y , where Y is the solution to (1.5).This turns out to be quite involved since we allow ρ to attain the value zero, and weagain want to work with equivalent changes of probability. We also make use of themonotonic limit theorem stated before (see section 1.1.1 or proposition 2.16). Detailsof this can be found in the first part of chapter 4 in Lemmas 4.1 and 4.2 and proposition4.3. Then establishing a comparison theorem for (1.5) is simply a combination of thislast result and the comparison theorem for simple BSDE 2.15

1.1.3 Randomization and constrained BSDE

In Kharroubi et al. [59] a technique called randomization of controls is introduced. Itconsists in substituting the control action with a random process taking values in theappropriate space and studying the dynamics of the state variable and this new processtogether. Paired to this randomization, a BSDE with a constraint on the martingale partof the solution is introduced and it turns out that its solution can be used to represent thevalue function of the original stochastic optimization problem. This has led to variousresults in optimal control, like for example a Feynman-Kac representation formula fornonlinear Hamilton Jacobi Bellman equations as stated in Kharroubi and Pham [58].More general details on the use of this approach can be found in the introduction tochapter 5. In the case of optimal stopping in a diffusive framework, its use has beenstudied in Fuhrman, Pham, and Zeni [41]. Here we extend this approach to our casewhere the probabilistic drivers are a marked point process measure and a Brownianmotion.

The starting point is the reflected BSDE introduced in the previous chapter, that werewrite here in the case with known generators

Yt = ξ +

∫ TtfsdAs +

∫ Ttgsds−

∫ Tt

∫EUs(y)q(dsdy)


Yt ≥ ht, ∀t ∈ [0, T ] a.s.∫ T0


1Yt−=ht−∀t ∈ [0, T ] a.s..

(1.7)

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We recall that q(dsde) = p(dsde) − φs(de)dAs is the compensated random mea-sure associated to a marked point process. This is defined on some probability space(Ω,F ,P) where we use the natural filtration generated by p and W . The solution liesin weighted L2,β spaces introduced in (1.1.1) and thoroughly defined in section 2.2. Wetake the same hypotheses we introduced in 1.1.2 that ensure existence and uniquenessof the solution. We recall also that the solution solves an optimal stopping problem,namely

Yt = ess supτ≥t

E[∫ τ∧T

t

fsdAs +

∫ τ∧T

t


∣∣∣∣ Ft] .What we do here is to randomize the control and substitute the choice of a stopping

time with a random one jump process. So instead of maximizing over all stoppingtimes, we introduce a “dual” problem in which we maximize over a family of changesof intensity (and thus of probabilities). We then show that the values of the two prob-lems is the same at time t = 0. To put this in place, we introduce a second probabilityspace (Ω′,F ′,P′) on which is defined a single jump point process Rt = 1η≤t, whereη ∼ exp(1) is an exponential random variable. The compensator of such process isDt = t ∧ η. We then construct an extended space (Ω, F , P) as the product space of(Ω,F ,P) and (Ω′,F ′,P′). On this new space p, W and R are independent. Here ap-pears a major difference to the purely diffusive case: we have to show that the compen-sator of p and R are still the same and that almost surely the jumps of p do not coincidewith the jump of R. This is done in lemma 5.1 with a monotone class argument.

We then introduce the constrained BSDE

Yt = ξ1η≥T +

∫ T

t

fs1[0,η]dAs +

∫ T

t

gs1[0,η]ds−∫ T

t

∫E

Us(e)q(dsde)

−∫ T

t

ZsdWs +

∫ T

t

hsdRs −∫ T

t

VsdRs + KT − Kt ∀t ∈ [0, T ],

(1.8)

with the constraintVt ≤ 0 dDt(ω)P(ω)− a.s. (1.9)

The spaces for the solution to this equation are the same as for the solution to thereflected BSDE (1.7), provided we build them over (Ω, F , P) and we add a similar onefor predictable processes integrable with respect to dD, the compensator of R. A solu-tion to this equation can be constructed directly from a solution to the reflected BSDE,as we prove in proposition 5.2. In particular, it holds that Yt(ω) = Yt(ω)1η(ω′)<t.Uniqueness in a classical ways does not hold, but we can find a minimal solution in thesense that the Y part of the solution is smaller than the one of any other solution.

This equation corresponds to a “dual control problem”, where we maximize the gainof the optimal stopping problem over a dominated family of probabilities. We introducethe set of functions

V =ν : Ω× [0,+∞)→ (0,+∞), F-predictable and bounded

.

Each ν ∈ V introduces a new probability Pν P through a Girsanov transform.Under Pν , W is still a Brownian motion, p still has compensator φs(de)dAs and R

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has compensator∫ t

0νsdDs, as shown in lemma 5.4. At this point we introduce the

following control problem

v(t) = ess supν∈V

Eν[ξ1η≥T +

∫ T∧η

t∧ηfsdAs +

∫ T∧η

t∧ηgsds+ hη1t<η<T

∣∣∣∣ Ft] .As we already announced, the solution to the constrained BSDE represents the valuefunction of this problem, that is

Yt = v(t).

The minimality property of Y is a consequence of this result. From the relation betweenY and Y this tells us that v(0) = v(0), i.e the normal control problem and the dual oneare the same at time zero. To prove this verification theorem for Y , we make use ofthe approximation result we established in the previous chapter. The idea is to createan approximating sequence Y n from Y starting from the approximating sequence forY , solution to the reflected BSDE. Then it is possible to prove that Y n solves a dualcontrol problem over a smaller space, and then pass to the limit.

This approach gives us another representation for the value function of the problem,through the use of constrained BSDE, offering another alternative to solve the problem.It is possible that algorithms based on this representation are more efficient than theircounterparts based on the reflected BSDE, but this has yet to be studied.

1.1.4 A switching problem for marked point processes

In chapter 6 we deal with an optimal switching problem for systems driven by markedpoint processes and Brownian motion. Again the problem is formulated in a nonMarkovian setting, so it will be solved with the help of BSDE. We place ourselvesagain in the setting we have been using, in a probability space (Ω,F ,P) that supports aBrownian motion W and a marked point process p with mark values in some measur-able space (E, E) and compensator φs(e)dAs. We fix a terminal time T . A completedescription can be found in 2.2.

We are given m modes of evolution, and we can switch from mode i to j facingsome cost Ct(i, j). To each mode there corresponds different running gains f i, gi anda terminal gain ξi. Changing the mode also modifies the law of the point process bymodifying its compensator in the following way. We are given m non-negative P ⊗ Efunctions ρi. Each ρi induces, through a Girsanov transform, a new probability Pi P. Under Pi, the compensator of p(dsde) becomes ρis(e)φs(de)dAs, thus effectivelychanging the dynamic of the point process. Although it introduces quite a numberof technical difficulties, it is important to let the ρi attain zero. This is useful if wewant to be able to treat the case with modes of evolution in which the probability ofhaving jumps is equal to zero. For this changes of probability to work we need to makeassumption (6-S), that we rewrite briefly here here:

• There exists a constant L > 0 such that for every i = 1, . . . ,m we have that0 ≤ ρit(e) ≤M .

• There exists a constant η > 3 +M4 such that E[eηAT

]<∞.

As in most switching problem, a strategy is a sequence of switching times θk andswitching actions αk for k ≥ 0, where θk are increasing stopping times and αk are Fθk-measurable random variables that take values in 1, . . . ,m. We say that a strategy is

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admissible if P(θk < T ∀k ≥ 0) = 0, and the set of admissible strategies starting fromtime t in mode i is denoted Ait.

Given a strategy a = (θk, αk)k≥0 ∈ Ait, we define the indicator of the current modeas

as =∑k≥1

αk−11(θk−1,θk](s)

and the “switched kernel” ρa by uniting the ρi in accordance to the strategy:

ρas =∑k≥1

1(θk−1,θk](s)ραk−1s (e). (1.10)

Through the use of ρa, each strategy a induces a probability measure Pa P. Wecan thus define the reward when we use strategy a ∈ Ait:

J(t, i, a) = Ea

[ξaT +

∫ T

t

fass dAs +

∫ T

t

gass ds−∑k≥1

Cθk(αk−1, αk)

∣∣∣∣∣Ft].

We then have the value function

v(t, i) = ess supa∈Ait

J(t, i, a).

To solve this problem, characterizing both the value function and the optimal strat-egy, we use a system of reflected BSDE with interconnected obstacles. This approachhas been introduced in Hu and Tang [53] and Hamadene and Zhang [50]. The idea isthat each equation corresponds to a mode and by combining them according to somestrategy, we can obtain the reward. In our case the system consist of m equations, onefor each mode, and has the form

Y it = ξi +

∫ Ttf isdAs +

∫ Ttgisds+

∫ Tt

∫EU is(e)(ρ

is(e)− 1)dAs

−∫ TtU is(e)q(dsde)−

∫ TtZisdWs +Ki

T −Kit

Y it ≥ max

j∈Ai(Y j

t − Ct(i, j))∫ T0

(Y it −max

j∈Ai(Y j

t − Ct(i, j)))dKit = 0.

(1.11)

Notice the presence of the integral of U i against ρi − 1. This term incorporates thefact that under Pi induced by ρi, the dynamic of the point process changes. Togetherwith the term−

∫ Tt

∫EU is(e)q(dsde) it forms a Pi martingale, which will then disappear

when we take expected value to obtain the expected reward. As we already said, thefact that the ρi can attain zero, and thus ρi − 1 can attain −1, introduce a series oftechnical complications when dealing with this system. In order to overcome them weneed to make an approximation to the compensator by adding ε > 0 to the ρi, obtain theneeded results under the probability induced by ρi + ε (equivalent to the reference one)and then send ε to zero. We do not detail them here and refer to the relevant chapter 6for the technical parts. As before, the solutions lie in the L2,β spaces we introduced in(1.1.1) and whose precise definition can be found in section 2.2.

For this equation to admit a solution and for the switching problem to make sense,we make assumption (6-D), that we rewrite here for convenience:

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Assumption : For i ∈ J and for some β > (M ′)2, where M ′ = max(M, |M − 1|)

i) ξi is a given FT measurable variable such that Ei[eβAT |ξi|2] <∞.

ii) f it is a progressive process in L2,β(A).

iii) git is a progressive process in L2,β(W ).

iv) Ct(i, j) for i, j ∈ J are continuous adapted processes. Ct(i, j) ≥ 0 for i 6= j andCt(i, i) = 0 and

inft

(Ct(i, j) + Ct(j, l)− Ct(i, l)) > 0 for all i 6= j 6= l.

Moreover, for all i, j ∈ J , E[supt e

βAtCt(i, j)]<∞.

v) It holds that for all i, j ∈ J ξi ≥ ξj − CT (i, j) a.s.

We recall that M is the bound for the ρi. Under this conditions we are able to provethe existence of a solution to the system. In order to do so we follow the Picard iterationmethod, which we briefly describe now.

The idea is to build a sequence of reflected BSDE where the obstacle is the maxi-mum of all the other BSDEs but at the previous step, of the form

Y i,nt = ξi +

∫ Ttf isdAs +

∫ Ttgisds+

∫ Tt

∫EU i,ns (e)(ρis(e)− 1)φs(de)dAs)

−∫ TtU i,ns (e)q(dsde)−

∫ TtZisdWs +Ki,n

T −Ki,nt

Y i,nt ≥ max

j∈Ai(Y i,n−1

t − Ct(i, j))∫ T0

(Y i,nt− −max

j∈Ai(Y i,n−1

t− − Ct(i, j)))dKi,nt = 0

(1.12)Thanks to some estimates and the comparison theorem 4.4 that we described at theend of section 1.4, we show that the solutions (Y i,n, U i,n, Zi,n, Ki,n) form a sequenceconverging to some limit (Y i, U i, Zi, Ki). These Y i satisfy the reflection condition butnot the minimal push condition. Thus we introduce another system of reflected BSDE,whose solution we denote Y i, where the obstacles are the maximum over the Y i. Wethen prove that Y i = Y i through comparisons and the use of the Snell envelope min-imality property, albeit under a different probability. Lastly we show that the processKi are continuous, which is important for the existence of an optimal strategy. Thetechnical details of this can be found in section 6.2.

The next step consist in proving the verification theorem 6.14. This states that Y it =

v(t, i) P-a.s., that is

Y it = ess sup

a∈AitEa

[ξaT +

∫ T

t

fass dAs +

∫ T

t

gass ds−∑k≥1

Cθk(αk−1, αk)

∣∣∣∣∣Ft].

We also identify an optimal strategy, which has the classical form “switch when you hit

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the barrier”:

θ∗n = inf

s ≥ θ∗n−1 : Y

α∗n−1s = max

j∈Aα∗n−1

(Y js − Cs(α∗n−1, j))

(1.13)

α∗n = arg maxj∈Aα∗n−1

(Y kθ∗n− Cθ∗n(α∗n−1, j)

)(1.14)

From this verification theorem we also obtain uniqueness of the Y i component and thusof (U i, Zi, Ki) by standard arguments, as stated in proposition 6.15.

This concludes the summary of the thesis. For convenience of reading, we recallonce more the contents of the various parts. In chapter 2 we introduce the mathemati-cal setting for the thesis, and state and prove the results about the non reflected BSDEas stated in section 1.1.1. In chapter 3 we study the reflected BSDE driven by a markedpoint process and Brownian motion we described in the first part of section 1.1.2, whilethe second part with an approximation result and a comparison theorem can be foundin chapter 4. Chapter 5 is dedicated to the randomization technique and constrainedBSDE we discussed in section 1.1.3. Lastly, in chapter 6 we treat the optimal switch-ing problem described in section 1.1.4, along with the system of reflected BSDE withinterconnected obstacles needed to solve it. In the last chapter 7 we draw conclusionsand make some comments about future work.

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CHAPTER2Mathematical tools

In the chapter we will first give an overview of the two main mathematical objectsthat are used in this thesis: marked point processes (MPP) and backward stochasticdifferential equations (BSDE). The former have been studied extensively in the past,and are now being used in conjunction with other new tools. The latter are more recent,starting with the paper Pardoux and Peng [66], and have known great popularity thanksfor their usefulness in stochastic optimal control, mathematical finance and PDE theory.

2.1 Marked point processes

We recall now some properties of marked point processes. For a comprehensive treat-ment of these type of processes, we refer the reader to Jacod [54], Bremaud [10] or Lastand Brandt [63]. Let (Ω,F ,P) be a complete probability space and let E be a Borelspace, i.e. a topological space homeomorphic to a Borel subset of a compact metricspace (sometimes called Lusin space; we recall that every separable complete metricspace is Borel). We call E the mark space and we denote by E its Borel σ-algebra. Apoint process is basically a sequence of random jump times and random “marks” in E,that is the places were the process jumps into.Definition 2.1. A marked point process (denoted MPP) is a sequence of random vari-ables (Tn, ξn)n≥0 with values in [0,+∞]× E such that P-a.s.

• T0 = 0.

• Tn ≤ Tn+1∀n ≥ 0.

• Tn <∞⇒ Tn < Tn+1∀n ≥ 0.

We will always assume the marked point process in the paper to be non-explosive,that is Tn → +∞ P-a.s. Another way of representing these processes is through the use

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Chapter 2. Mathematical tools

of an integer random measure. To each marked point process we associate a randomdiscrete measure p on ((0,+∞)× E,B((0,+∞))⊗ E):

p(ω,D) =∑n≥1

1(Tn(ω),ξn(ω))∈D,

We refer to p also as marked point process. Such measure carries the same informationas the sequence (Tn, ξn)n≥0. For each C ∈ E , define the counting process Nt(C) =p((0, t] × C) that counts how many jumps have occurred to C up to time t. DenoteNt = Nt(E). They are right continuous increasing process starting from zero. Eachpoint process generates a filtration G = (Gt)t≥0 as follows: define for t ≥ 0

G0t = σ(Ns(C) : s ∈ [0, t], C ∈ E)

and set Gt = σ(G0t ,N ), where N is the family of P-null sets of F . G is a right-

continuous filtration that satisfies the usual hypotheses. Denote by PG the σ-algebra ofG-predictable processes.

For each marked point process there exists a unique predictable random measureν, called compensator, such that for all non-negative PG ⊗ E-measurable process C itholds that

E[∫ +∞

0

∫E

Ct(e)p(dtde)

]= E

[∫ +∞

0

∫E

Ct(e)ν(dtde)

].

Similarly, there exists a unique right continuous increasing process with A0 = 0, thedual predictable projection of N , such that for all non-negative predictable processesD

E[∫ +∞

0

DtdNt

]= E

[∫ +∞

0

DtdAt

].

It is possible to choose a version of ν that satisfies identically

ν(t × E) ≤ 1 (2.1)

It is known that, thanks to the assumptions on the space E, there exists a function φon Ω×[0,+∞)×E such that we have the disintegration ν(ω, dtde) = φt(ω, de)dAt(ω).Moreover the following properties hold:

• for every ω ∈ Ω, t ∈ [0,+∞), C 7→ φt(ω,C) is a probability on (E, E).

• for every C ∈ E , the process φt(C) is predictable.

Unless otherwise stated, all marked point processes in this thesis have a compensatorof this form.

2.1.1 Integrals with respect to marked point processes

From now on, fix a terminal time T > 0. Next we need to define integrals with re-spect to point processes. Define first the compensated measure q(dtde) = p(dtde) −φt(de)dAt. This is a martingale measure in the sense that for any F ∈ E we have thatthe process

q((0, t]× F )

is a martingale.

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2.1. Marked point processes

Definition 2.2. Let C be a PG ⊗ E-measurable process such that∫ T

0

∫E

|Ct(e)|φt(de)dAt <∞.

Then we can define the integral∫ T

0

∫E

Ct(e)q(dtde) =

∫ T

0

∫E

Ct(e)p(dtde)−∫ T

0

∫E

Ct(e)φt(de)dAt

as difference of ordinary integrals with respect to p and φdA.

Remark 2.3. In the paper we adopt the convention that∫ ba

denotes an integral on (a, b]if b <∞, or on (a, b) if b =∞.

Remark 2.4. Since p is a discrete random measure, the integral with respect to p is asum: ∫ t

0

∫E

Cs(e)p(dsde) =∑Tn≤t

CTn(ξn)

We have the following result

Proposition 2.5: Let C be a predictable random field that satisfies

E[∫ T

0

∫E

|Ct(e)|φt(de)dAt]<∞. (2.2)

Then the integral ∫ t

0

∫E

Cs(e)q(dsde) (2.3)

is a martingale.

Notice that this needs not to be a martingale in L2. We have the following result:

Proposition 2.6: Assume that, in addition to (2.2), the predictable random field C sat-isfies

E[∫ T

0

∫E

|Cs|2φs(de)dAs]<∞.

Then the martingale Mt = M0 +∫ t

0

∫ECsq(dsde) is square integrable.

Proof. Consider the dynamic of M2t :

M2t = M2

0 + 2

∫ t

0

∫E

Ms−dMs +∑

0<Tn≤t

(∆MTn)2

= M20 + 2

∫ t

0

∫E

Ms−Cs(e)q(dsde) +∑

0<Tn≤t

CTn(ξn)2

Consider now the sequence of stopping times

τn = inft > 0 : Mt > n.

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It is clear that τn +∞. Then between 0 and τn we have

E [Mτn ] = E[M2

0

]+ 2E

[∫ τn

0

∫E

Ms−Cs(e)q(dsde)

]+ E

[ ∑0<Tn≤τn

CTn(ξn)2

]

= E[M2

0

]+ E

[ ∑0<Tn≤τn

CTn(ξn)2

]

≤ E[M2

0

]+ E

[ ∑0<Tn≤T

CTn(ξn)2

].

We conclude by taking the limit and using Fatou’s lemma.

The following assumption will be used through all the thesis

Assumption 2-A: The process A is continuous.

It is equivalent (see He, Wang, and Yan [52, Corollary 5.28]) to stating that theprocess N is quasi-left continuous, i.e. it has only totally inaccessible jumps. Thisentails that the marked point process p and all the martingales of the form (2.3) havetotally inaccessible jumps.

We also have a characterization of the square and predictable bracket of this class ofintegrals.

Proposition 2.7: Let M be a martingale of the form

Mt = M0 +

∫ t

0

∫E

Cs(e)q(dsde)

with C such that

E[∫ T

0

∫E

|Cs(e)|2φs(de)dAs]<∞.

Then we have

[M ]t =

∫ t

0

∫E

C2s (e)p(dsde) =

∑0<Tn≤t

C2Tn(ξn)

〈M〉t =

∫ t

0

∫E

C2s (e)φs(de)dAs

Another important ingredient is the martingale representation theorem:

Theorem 2.8: Let M be a right continuous process adapted to G. M is a uniformlyintegrable martingale if and only if there exists a P ⊗ E-measurable process U , withE[∫ T

0

∫E|Us(e)|φs(de)dAs

]<∞, such that

Mt = M0 +

∫ t

0

∫E

Us(e)q(dsde).

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2.1. Marked point processes

2.1.2 Changes of probability

As we have seen for a fixed probability there is one compensator. The converse is alsotrue: given a predictable measure η on R+ × E then there exists a probability underwhich the compensator of the point process is η. There are two forms of this problem,both introduced in Jacod [54]. One deals with the case where a predictable randommeasure is given, there is no reference probability and is set on the space of “canonicalpaths”. This is not particularly useful in this work, so we refer the reader to the originalarticle. The other form is when there is a reference probability, and one does a changeof probability “a la Girsanov”: a new probability, absolutely continuous with respect tothe reference one, is introduced through an exponential martingale. In this case there isa characterization of the compensator under the new probability, which we will detailin the following.

Again we work in the setting from before. This can be done without the continuityassumption on the process A, but since we will use it our work there is no need tointroduce the complicated notation for the general case.

The basic ingredient here is what we will call the kernel, a non-negative P ⊗ Efunction ρ. Consider the process

Ht =

∫ t

0

∫E

(ρs(e)− 1)q(dsde)

and its “Doleans-Dade” exponential Z:

Lt = E(ρ)t =∏

0<Tn≤t

ρTn(ξn)e∫ t0

∫E(1−ρs(e))φs(de)dAs . (2.4)

Lt is a positive supermartingale and a local martingale. When L is a martingale itintroduces a probability Pρ P defined as

dPρ

dP= LT .

Under this new probability, the compensator of the point process p is given by

ρs(e)φs(de)dAs.

We will denote by qρ(dsde) = p(dsde) − ρs(e)φs(de)dAs the compensated measureunder Pρ.

In the diffusive case there are several conditions under which L is a martingale. A(quite strong) condition is to ask for the Girsanov kernel being bounded. In the caseof marked point processes this is not enough, and we need to impose an integrabilitycondition on the process A. The one here is used in Confortola and Fuhrman [19],which is in turn based on Bremaud [10, T11,Chapter VIII].

Assumption (2-K): There exist M > 0 such that for all i ∈ J

0 ≤ ρit(e) ≤M (2.5)

We also need the following assumption to use changes of probabilities

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Assumption (2-K’): For some η such that

η > 3 +M4 (2.6)

it holds thatE[eηAT

]<∞. (2.7)

In the result we also add another property of L that will be used in the following.

Lemma 2.9: Under assumptions (2-K) and (2-K’) L is a square integrable martingale.Moreover it holds that

E [(Lt)2 |Fs]

(Ls)2< E

[eηAT

∣∣ Ft] P-a.s. (2.8)

Proof. The proof of the first part can be found in Confortola and Fuhrman [19, Lemma4.2]. The authors prove that if for some γ > 1 and

β = γ + 1 +Mγ2

γ − 1

it holds that E[eβAT

]< ∞, then supt∈[0,T ] E [Lγt ] < ∞ and E [LT ] = 1. We don’t

repeat the proof here, but mention that the proof still holds if we choose β to be anynumber strictly greater than γ + 1 + Mγ2

γ−1. In particular the statement of our theorem

refers to the case γ = 2. As for the proof of the inequality (2.8) we have (which is ageneralization of what done for the first part of the lemma). The quantity

E [(Lt)2 |Fs]

(Ls)2

can be rewritten as

E

∏

0<Tn≤tρ2Tn

(ξn)∏0<Tn≤s

ρ2Tn

(ξn)exp

1

2

∫ t

s

∫E

(1− (ρs(e))

4)φs(de)dAs

·

exp

2

∫ t

s

∫E

(1− ρs(e))φs(de)dAs −1

2

∫ t

s

∫E

(1− (ρs(e))

4)φs(de)dAs

∣∣∣∣∣∣∣Fs .

Using Cauchy-Schwarz inequality for conditional expectations, this becomes

E[exp

4

∫ t

s

∫E

(1− ρs(e))φs(de)dAs −∫ t

s

∫E

(1− (ρs(e))

4)φs(de)dAs

∣∣∣∣Fs]1/2

E[E(ρ4)tE(ρ4)s

∣∣∣∣Fs]1/2

.

The last term is less or equal than 1, thanks to the supermartingale property of E(ρ4),and the other term is easily estimated:

E[exp

∫ t

s

∫E

(3− 4ρs(e))φs(de)dAs +

∫ t

s

∫E

(ρs(e))4φs(de)dAs

∣∣∣∣ Fs]≤ E

[exp

(3 +M4)(At − As)

∣∣ Fs] < E[eηAT

∣∣ Fs] .18

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2.2. Probabilistic setting

2.2 Probabilistic setting

In addition to the marked point process we just introduced, we will also have a diffusivecomponent. Unless otherwise stated, the setting described here will be used through thepaper.

Let (Ω,F ,P) be a complete probability space and p(dtde) a non-explosive markedpoint process on a Borel space (E, E) as before, whose compensator is φt(de)dAt. LetT > 0 be a fixed terminal time. In addition we assume we are given an independentWiener process W in Rd. Let G = (Gt)t≥0 (resp. F = (Ft)t≥0) be the completedfiltration generated by p (resp. p andW ), which satisfies the usual conditions. Let Tt bethe set of F-stopping times greater than t. Denote by P (resp. Prog) be the predictable(resp. progressive) σ-algebra relative to F. As previously said, we will assume theprocess A to be continuous.

For some parameter β > 0, we introduce the following spaces:

• Lr,β(A) (resp. Lr,β(A,G)) is the space of all F-progressive (resp. G-progressive)processes X such that

||X||rLr,β(A) = E[∫ T

0

eβAs|Xs|rdAs]<∞.

• Lr,β(p) (resp. Lr,β(p,G)) is the space of all F-predictable (resp. G-predictable)processes U such that

||U ||rLr,β(p) = E[∫ T

0

∫E

eβAs|Us(e)|rφs(de)dAs]<∞.

• Lr,β(W,Rd) (resp. Lr,β(W,Rd,G)) is the space of F-progressive (resp. G-progressive)processes Z in Rd such that

||Z||rLr,β(W ) = E[∫ T

0

eβAs|Zs|rds]<∞

• I2 (resp. I2(G)) is the space of all cadlag increasing F-predictable (resp. G-predictable) processes K such that E[K2

T ] <∞.

We indicate by Lr,β(A), Lr,β(p), Lr,β(W ) and I2 the respective space of equivalenceclasses.

Notice that L2,β(p) ⊂ L1,0(p) thanks to Holder inequality. Indeed we have that fora process C ∈ L2,β(p)

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E[∫ T

0

∫E

|Cs(e)|φs(de)dAs]

= E[∫ T

0

∫E

e−β2Ase

β2As|Cs(e)|φs(de)dAs

]≤ E

[(∫ T

0

e−βAsdAs

)1/2(∫ T

0

∫E

eβAs|Cs(e)|2φs(de)dAs)1/2

]

= E

(∣∣∣∣−e−βAsβ

∣∣∣∣T0

)1/2(∫ T

0

∫E


= E

[(1− eβAt

β

)1/2(∫ T

0

∫E


]≤ 1

β1/2||C||L2,β(p).

The trick used above will be often used in the thesis.In this framework of marked point process and Brownian motion, the martingale

representation theorem still holds:

Theorem 2.10: Let M be a right continuous square integrable F-martingale. Then hereexists a process U ∈ L1,0(p) and a process Z ∈ L2,0(W ), such that

Mt = M0 +

∫ t

0

∫E

Us(e)q(dsde) +

∫ t

0

ZsdWs.

This is stated in Becherer [5] and can be shown similarly to Cohen and Elliott [16,Theorem 14.5.7].

Changes of probability of Grisanov type can also be done. To this end we usetwo kernels: ρ relative to the point process as described in section 2.1.2, and for theBrownian part a progressive process θ with values in Rd and such that |θt| < K forsome constant K. It is possible to use weaker assumptions for θ, but we do not needsuch generality in the present work. Notice that if θ ≡ 0, then the change of probabilitychanges the compensator of p leaving W a Brownian motion.

Lemma 2.11: Let ρ be a P ⊗ E-measurable function and θ a progressive process. As-sume ρ and A satisfy assumptions (2-K) and (2-K’) for some M and η, and assume that|θt| < K for some constant K. Then the martingale

Lρ,θt =∏

0<Tn≤t

ρTn(ξn)e∫ t0

∫E(1−ρs(e))φs(de)dAs−

∫ t0 |θs|

2ds+∫ t0 θsdWs

induces a new probability Pρ,θ P defined as

dPρ,θ

dP= Lρ,θT .

Under Pρ,θ, W −∫ t

0θsds is a Brownian motion and the compensator of p(dsde) is

ρs(e)φs(de)dAs. Moreover it holds that

E[(Lρ,θT )2

∣∣∣ Ft](Lρ,θt )2

< E[eηAt

∣∣ Ft]20

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2.2. Probabilistic setting

Proof. Girsanov theorem for the Brownian motion is well known. In particular it iseasy to see that if we consider the change induced by the θ part alone, it does notchange the compensator of the point process. Indeed if Pθ is the probability induced byθ, then for a non negative P ⊗E-measurable process C we have, using Dellacherie [23,T47,Chapter 4],

Eθ[∫ T

0

∫E

Cs(e)p(dsde)

]= E

[LθT

∫ T

0

∫E

Cs(e)p(dsde)

]= E

[∫ T

0

∫E

LθsCs(e)p(dsde)

]= E

[∫ T

0

∫E

LθsCs(e)φs(de)dAs

]= E

[LθT

∫ T

0

∫E

Cs(e)φs(de)dAs

]= Eθ

[∫ T

0

∫E

Cs(e)φs(de)dAs

].

Similarly, if we change through a process ρ, W is still a Brownian motion. So what wecan do is introduce Pθ as

dPθ

dP= LθT

and introduce Pρ,θ asdPρ,θ

dPθ= LρT .

In this case the passage martingale from P is Lρ,θt . Under Pρ,θ, Wt −∫ t

0θsds is a

Brownian motion and the compensator of p(dsde) is ρs(e)φs(de)dAs. The proof that

E[(Lρ,θT )2

∣∣∣ Ft](Lρ,θt )2

< E[eηAt

∣∣ Ft]is the same as in lemma 2.9. We just have to observe that

(Lθt )2 = e2

∫ t0 θsdWs−

∫ t0 |θs|

2ds

= e3∫ t0 |θs|

2dse−14

∫ t0 16|θs|2ds+ 1

2

∫ t0 4θsdWs = e3

∫ t0 |θs|

2ds√L4θt

and the second piece, together with the point process exponential supermartingale,forms the supermartingale Lρ

4,4θt when squared. The

e3∫ t0 |θs|

2ds ≤ e3K2T

is bounded and poses no problem.

Lemma 2.12: Le the conditions on lemma 2.11 hold. Let M be a P-martignale of theform

Mt = M0 +

∫ t

0

∫E

Us(e)q(dsde) +

∫ t

0

ZsdWs

with U ∈ L2,0(p) and a process Z ∈ L2,0(W ). then the process M defined by

Mt = M0 +

∫ t

0

∫E

Us(e)(p(dsde)− ρs(e)φs(de)dAs) +

∫ t

0

Zs(dWs − θsds)

is a Pρ,θ martingale.

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Proof. Clearly ∫ t

0

Zs(dWs − θsds)

is a local Pρ,θ-martingale. By using Burkholder-Davis-Gundy inequality we have that

Eρ,θ[

supt∈[0,T ]

∣∣∣∣∫ t

0

Zs(dWs − θsds)∣∣∣∣]≤ CEρ,θ

[(∫ T

0

|Zs|2ds) 1

2

]

= CE

[Lρ,θT

(∫ T

0

|Zs|2ds) 1

2

]

≤ CE[(Lρ,θT )2

] 12 E[∫ T

0

|Zs|2ds] 1

2

<∞.

As for the point process part we do the same

Eρ,θ[

supt∈[0,T ]

∣∣∣∣∫ t

0

∫E

Us(e)qρ(dsde)

∣∣∣∣]≤ KEρ,θ

[(∫ T

0

∫E

|Us(e)|2p(dsde))1/2

]

= KE

[Lρ,θT

(∫ T

0

∫E

|Us(e)|2p(dsde)1/2

]

≤ KE[(Lρ,θT )2

] 12 E[∫ T

0

∫E

|Us(e)|2φs(de)dAs] 1

2

<∞.

2.3 Backward stochastic differential equations

We have already talked a bit about the story of BSDE in the introduction. In this sectionwe give a brief review of the theory of these equations, concentrating ourselves onBSDE driven by a marked point process. We also give a couple of results when thereis an added Brownian term, in particular we state a comparison theorem. A generalintroduction to BSDE can be found for example in El Karoui, Peng, and Quenez [29],Pardoux and Rascanu [68], Crepey [21], Pardoux [67].

The simplest BSDE (in the Brownian setting) is the following

Yt = ξ +

∫ T

t

f(s, Ys, Zs)ds−∫ T

t

ZsdWs.

or in differential form −dYt = f(t, Yt, Zt)− ZtdWt

YT = ξ

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2.3. Backward stochastic differential equations

Basically a BSDE is a stochastic equation were a terminal data ξ is given, along aLipschitz function f , called the generator, defining the dynamic. The idea is that thisequation is solved backwards in time, but using the theory of process adapted to afiltration. For this reason the there are two unknowns (Y, Z) in a BSDE: Y whichis the main part of the solution and a martingale part (given by the integral of theprocess Z) that allows for the solution to be adapted. The key ingredient in this fieldis the martingale representation theorem. In the classical diffusive case this is typicallysolved when ξ and f are square integrable and f is Lipschitz in (Y, Z).

We are interested in the case of BSDE driven by marked point process (or markedpoint process and Brownian motion). This class of equations was studied in Confortolaand Fuhrman [19] and Confortola, Fuhrman, and Jacod [20] and has the form

Yt = ξ +

∫ T

t

f(s, Ys, Us)dAs −∫ T

t

∫E

Us(e)q(dsde), (2.9)

where the unknowns are (Y, U). Notice that the generator is integrated not withrespect to the Lebesgue measure, but rather with respect to the random measure dA.(2.9) is solved provided the data is square integrable in some sense and the generator isLipschitz. To be precise it must be assumed that the process A is continuous and thatfor some β > 0

Assumption (2-N): 1. The final condition ξ : Ω→ R is GT measurable and

E[eβAT |ξ|2

]<∞.

2. For every ω ∈ Ω, t ∈ [0, T ], r ∈ R, the mapping ft(ω, r, ·) : L2(E, E , φt(ω, de))→R satisfies the assumptions:

(i) for every U ∈ L2,β(p) the mapping

(ω, t, r) 7→ ft(ω, r, Ut(ω, ·))

is ProgG ⊗ B(R)-measurable;

(ii) there exists Lf ≥ 0 and LU ≥ 0 such that for every ω ∈ Ω, t ∈ [0, T ],r, r′ ∈ R, z, z′ ∈ L2(E, E , φt(ω, de)) we have

|ft(ω, r, z(·))− ft(ω, r′, z′(·))|

≤ Lf |r − r′|+ LU

(∫E

|z(e)− z′(e)|2φt(ω, de))2

;

(iii) we have

E[∫ T

0

eβAt |ft(0, 0)|2dAt]<∞.

then the following existence and uniqueness theorem holds:

Theorem 2.13: Suppose that the hypotheses above hold for some β > 2Lf +L2U . Then

there exists a unique solution to equation (2.9) in L2,β(A)× L2,β(p).

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In Confortola, Fuhrman, and Jacod [20] the same equation is solved in a L1 setting,and in Confortola [18] in an Lp setting. In all these cases the process A is asked to becontinuous.

It is possible to add a Wiener motion to the framework, as we will do now and laterin chapter 3. In that case a second generator integrated with respect to the Lebesguemeasure in time is added:

Yt = ξ +

∫ T

t

fs(Ys, Us)dAs +

∫ T

t

gs(Ys, Zs)ds−∫ T

t

∫E


t

ZsdWs.

(2.10)The hypotheses used are the same as the ones above, provided we use F as filtration

and we add the following assumptions on g

Assumption (2-N’):

(i) g is Prog × B(R)× B(Rd)-measurable.

(ii) There exist Lg ≥ 0, LZ ≥ 0 such that for every ω ∈ Ω, t ∈ [0, T ], y, y′ ∈ R,z, z′ ∈ Rd

|g(ω, t, y, z)− g(ω, t, y′, z′)| ≤ Lg|y − y′|+ LZ |z − z′|

(iii) we have

E[∫ T

0

eβAs|g(s, 0, 0)|2ds]<∞.

Then we have

Theorem 2.14: Suppose that the hypotheses above hold for some β > 2Lf +L2U . Then

there exists a unique solution to equation (2.10) (L2,β(A) ∩ L2,β(W )) × L2,β(p) ×L2,β(W ).

The proof we give here will be schematic, as the result is a straightforward extensionof the case of only a marked point process involved, and the idea of the proof is coveredin the slightly more complicated case of the reflected BSDE in chapter 3 in proposition3.6 and theorem 3.17.

Proof. First consider the case when f, g are known processes in L2,β(A) and L2,β(W )respectively, for some β > 0. We define the square integrable martingale

Mt = E[ξ +

∫ T

0

fsdAs +

∫ T

0

gsds

∣∣∣∣ Ft]and the process

Yt = Mt −∫ t

0

fsdAs −∫ t

0

gsds.

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By the martingale representation theorem there exists U and Z such that Mt = M0 +∫ t0

∫EUs(e)q(dsde) +

∫ t0ZsdWs. Then all these processes satisfy

Yt = ξ +

∫ T

t

fsdAs +

∫ T

t

gsds−∫ T

t

∫E


t

ZsdWs.

Since Y satisfies

Yt = E[ξ +

∫ T

t

fsdAs +

∫ T

t

gsds

∣∣∣∣ Ft] ,and the data is in the L2,β spaces we can deduce (see lemma 3.7) that

E supt∈[0,T ]

eβAs|Ys|2.

Define the sequence of stopping times

Sn = inf

t ∈ [0, T ] :

∫ t

0

eβAs|Ys|2dAs +

∫ t

0

∫E

eβAs|Us(e)|2φs(de)dAs

+

∫ t

0

eβAs|Zs|2ds > n

,

and consider the “Ito Formula” applied to eβ(At+t)Y 2t between 0 and Sn. We obtain

eβ(ASn+Sn)Y 2Sn ≥ β

∫ Sn

0

eβ(As+s)Y 2s dAs + β

∫ Sn

0

eβ(As+s)Y 2s ds

+ 2

∫ Sn

0

eβ(As+s)Ys−Us(e)q(dsde) + 2

∫ Sn

0

eβ(As+s)YsZsdWs

− 2

∫ Sn

0

eβ(As+s)YsfsdAs − 2

∫ Sn

0

eβ(As+s)Ysgsds

+

∫ Sn

0

∫E

eβ(As+s)U2s (e)φs(de)dAs +

∫ Sn

0

eβ(As+s)Z2sds

+

∫ Sn

0

∫E

eβ(As+s)U2s (e)q(dsde).

The integrals in q(dsde) and dW , thanks to the definition of Sn, are martingales. Thiscan be proven by direct manipulation and the use of Burkholder Davis Gundy inequal-ity. After expectation, use of Young’s inequality on the integrals with Ysfs and Ysgsand reordering we obtain

β

2E[∫ Sn

0

eβ(As+s)Y 2s dAs

]+ E

[∫ Sn

0

∫E

eβ(As+s)U2s (e)φs(de)dAs

]+β

2E[∫ Sn

0

eβ(As+s)Y 2s ds

]+ E

[∫ Sn

0

eβ(As+s)Z2sds

]≤ E

[supteβ(At+t)Y 2

t

]+

2

βE[∫ T

0

eβ(As+s)f 2s dAs

]+

2

βE[∫ T

0

eβ(As+s)g2sds

]25

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Now let S = limn Sn and by the last estimate, considering how Sn are defined, we haveS = T . This implies that Y ∈ L2,β(A) ∩ L2,β(W ), Z ∈ L2,β(W ) and U ∈ L2,β(p).

We now consider the case where f and g depend on (Y, U) and (Y, Z) respectively.We need to consider the space L2,β(A+ λ) of all processes Y such that

‖Y ‖2L2,β(A+λ) = E

[∫ T

0

eβAsY 2s (dAs + ds)

]<∞.

Notice that this is equivalent to the norm we used on L2,β(A) ∩ L2,β(W ). From thefirst part of the proof, we can associate to each triple (P,Q,R) ∈ L2,β(A + λ) ×L2,β(p)×L2,β(W ) a triple (Y, U, Z) ∈ L2,β(A+λ)×L2,β(p)×L2,β(W ) solution to theBSDE with generators fs(Ps, Qs) and gs(Ps, Rs), thus we have defined a map Φ fromL2,β(A+λ)×L2,β(p)×L2,β(W ) to itself. Let (Y , U , Z) = (Y ′, U ′, Z ′)−(Y ′′, U ′′, Z ′′)and (P , Q, R) = (P ′, Q′, R′) − (P ′′, Q′′, R′′), where (Y ′, U ′, Z ′) = Φ((P ′, Q′, R′))and (Y ′′, U ′′, Z ′′) = Φ((P ′′, Q′′, R′′)). By applying Ito’s formula to eβAseγs|Ys|2 andperforming computations as in theorem 3.17, we obtain for some α > 0(

β −L2p

α− Lf√

α

)||Y ||2β,γ,A + ||U ||2β,γ,p

+

(γ − L2

W

α− Lg√

α

)||Y ||2β,γ,W + ||Z||2β,γ,W

≤ Lf√α||P ||2β,γ,A + α||Q||2β,γ,p + Lg

√α||P ||2β,γ,W + α||R||2β,γ,W .

In the above expression ||X||2β,γ,A denotes the norm

||X||2β,γ,A = E[∫ T

0

eβAseγs|Xs|2dAs].

||X||2β,γ,p and ||X||2β,γ,W are defined similarly. Since β > L2p + 2Lf , it is possible to

chose α ∈ (0, 1) such that

β >L2p

α+

2Lf√α,

leading to

Lf√α||Y ||2β,γ,A +

Lg√α||Y ||2β,γ,W + ||U ||2β,γ,p + ||Z||2β,γ,W

≤ α

(Lf√α||P ||2β,γ,A + ||Q||2β,γ,p +

Lg√α||P ||2β,γ,W + ||R||2β,γ,W

),

Where γ was chosen large enough. This tells us that the map Φ is a contraction for theequivalent norm

E[∫ T

0

∫E

eβAseγs|Us(e)|2φs(de)dAs]

+ E[∫ T

0

eβAseγs|Zs|2ds]

+ E[∫ T

0

eβAseγs|Ys|2(Lf√αdAs +

Lg√αds

)].

Then there exist one and only one fixed point for Φ, and the theorem is proven.

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One important tool in BSDE theory is the comparison theorem, that allows to estab-lish inequality relations on the solutions based on inequalities on the data. In generalBSDE with jumps component require additional assumptions for this result to hold (seeBarles, Buckdahn, and Pardoux [4] for a counterexample), and BSDE driven markedpoint processes are no exception. There are a number of works that provide a com-parison principle for BSDE with jumps, and we cite among others Cohen, Elliott, andPearce [17], Kruse and Popier [61], Quenez and Sulem [73], and Royer [77]. We givehere a result that will be useful in our case, for the BSDE (2.10). Compared to the caseof point processes that have a compensator absolutely continuous with respect to theLebesgue measure, we need to add some integrability condition on the process A sincewe will be using Girsanov changes of measure.

Theorem 2.15: Let (ξi, f i, gi)i=1,2 be two sets of data for which the hypotheses abovehold. Let (Y i, U i, Zi)i=1,2 be the corresponding solutions. Assume that ξ2 ≤ ξ1 a.s,f 2t (Y 1

t , U1t ) ≤ f 1

t (Y 1t , U

1t ) and g2

t (Y1t , Z

1t ) ≤ g1

t (Y1t , Z

1t ) a.s. for all t. Assume there is

a P ⊗ E-measurable function γ such that −1 ≤ γ ≤ C and for all t, y ∈ R+ × R andu2, u1 ∈ L2(E, E , φt(ω, de))

f 2t (y, u2)− f 2

t (y, u1) ≤∫E

γt(e)(u2(e)− u1(e))φt(de). (2.11)

Assume also that for some η > 3 + (C + 1)4 we have E[eηAT

]< ∞. Then we have

that Y 2t ≤ Y 1

t a.s. for all t.

Proof. To simplify notation suppose that the Brownian motion is in d = 1, the generalcase is done as in the case with only a diffusion part. Define Y = Y 2 − Y 1, U =U2 − U1, Z = Z2 − Z1, ξ = ξ2 − ξ1, f = f 2(Y 1, U1) − f 1(Y 1, U1) and g =g2(Y 1, Z1)− g1(Y 1, Z1). Y satisfies

Yt = ξ +

∫ T

t

(f 2s (Y 2

s , U2s )− f 1

s (Y 1s , U

1s ))dAs +

∫ T

t

(g2s(Y

2s , Z

2s )− g1

s(Y1s , Z

1s ))dAs

−∫ T

t

∫E


t

ZsdWs (2.12)

Define the quantities

αs =f 2s (Y 2

s , U2s )− f 2

s (Y 1s , U

2s )

Ys1Ys 6=0 ≤ Lf

βs =g2s(Y

2s , Z

2s )− g2

s(Y1s , Z

2s )

Ys1Ys 6=0 ≤ Lg

θs =g2s(Y

1s , Z

2s )− g2

s(Y1s , Z

1s )

Zs1Zs 6=0 ≤ LZ .

The equation can be thus rewritten as

Yt = ξ +

∫ T

t

αsYsdAs +

∫ T

t

(f 2s (Y 1

s , U2s )− f 2

s (Y 1s , U

1s ))dAs +

∫ T

t

fsdAs

+

∫ T

t

βsYsds+

∫ T

t

θsZsds+

∫ T

t

gsds−∫ T

t

∫E


t

ZsdWs.

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Consider now the positive process Γ solution todΓt = αtdAt + βtdt

Γ0 = 1.(2.13)

We have thatΓt = e

∫ t0 αsdAse

∫ t0 βsds = BtCt,

where Bt = exp∫ t

0αsdAs

and Ct = exp

∫ t0βsds

< eLgT . We notice that

B2t ≤ e2LfAt < eβAt . (2.14)

We consider now the dynamic of the product Y Γ. We obtain

ΓT ξ = ΓtYt +

∫ T

t

αsYsΓsdAs +

∫ T

t

βsYsΓsds−∫ T

t

αsYsΓsdAs

−∫ T

t

Γs(f2s (Y 1

s , U2s )− f 2

s (Y 1s , U

1s ))dAs −

∫ T

t

ΓsfsdAs

−∫ T

t

βsYsΓsds−∫ T

t

θsZsΓsds−∫ T

t

Γsgsds

+

∫ T

t

∫E

ΓsUs(e)q(dsde) +

∫ T

t

ΓsZsdWs.

Remembering that ξ, f and g are non-positive while Γ is non-negative we obtain

YtΓt ≤∫ T

t

Γs(f2s (Y 1

s , U2s )− f 2

s (Y 1s , U

1s ))dAs +

∫ T

t

θsZsΓsds

−∫ T

t

∫E

ΓsUs(e)q(dsde)−∫ T

t

ΓsZsdWs.

Using the condition (2.11) the last inequality becomes

YtΓt ≤∫ T

t

∫E

γs(e)ΓsUs(e)φs(e)dAs +

∫ T

t

θsZsΓsds

−∫ T

t

∫E

ΓsUs(e)q(dsde)−∫ T

t

ΓsZsdWs. (2.15)

We have that ΓU ∈ L2,0(p) and ΓZ ∈ L2,0(W ) and the terms in q(dsde) and dWare martingales, indeed

E[∫ T

0

∫E

|ΓsUs(e)|2φs(de)dAs]< E

[∫ T

0

∫E

eβAs|Us(e)|2φs(de)dAs]<∞

E[∫ T

0

|ΓsZs|2ds]< E

[∫ T

0

eβAs|Zs|2ds]<∞

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Now reorder the terms in (2.15) to obtain

YtΓt ≤ −∫ T

t

ΓsZs(dWs − θsds)

−∫ T

t

∫E

ΓsUs(e)(p(dsde)− (γs(e) + 1)φs(de)dAs.

Consider ε such that η > 3 + (C + 1 + ε)4. Then for ε < ε we add to both sides of theprevious inequality the term

ε

∫ T

t

ΓsUs(e)φs(de)dAs,

obtaining

YtΓt + ε

∫ T

t

ΓsUs(e)φs(de)dAs ≤ −∫ T

t

ΓsZs(dWs − θsds)

−∫ T

t

∫E

ΓsUs(e)(p(dsde)− (γs(e) + 1 + ε)φs(de)dAs. (2.16)

Now we can consider γs(e) + 1 + ε (which satisfies, together with the conditionon A in the statement of the theorem, assumptions (2-K) and (2-K’)) and θs (which isbounded) as Girsanov kernels. We can introduce the probability Pγ,θ,ε ∼ P through theexponential martingale here denoted Lγ,θ,ε. It is equivalent since the part relative to thepoint process of Girsanov kernel is strictly positive. The P-martingales above are Pγ,θ,εmartingales, thanks to 2.12, and thus by taking Pγ,θ,ε expectation conditional on Ft in(2.16) they vanish. Thus for any ε < ε we have that

YtΓt + εEγ,θ,ε[∫ T

t

ΓsUs(e)φs(de)dAs

∣∣∣∣ Ft] ≤ 0 Pγ,θ,ε-a.s.

and thus

YtΓt + εEγ,θ,ε[∫ T

t

ΓsUs(e)φs(de)dAs

∣∣∣∣ Ft] ≤ 0 P-a.s.

since they are equivalent. To conclude the theorem we just have to show that

εEγ,θ,ε[∫ T

t

ΓsUs(e)φs(de)dAs

∣∣∣∣ Ft]→ 0. (2.17)

We have that

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Eγ,θ,ε[∫ T

t

ΓsUs(e)φs(de)dAs

∣∣∣∣ Ft] ≤ Eγ,θ,ε[∫ T

0

Γs|Us(e)|φs(de)dAs∣∣∣∣ Ft]

≤ eLgTEγ,θ,ε[∫ T

0

Bs|Us(e)|φs(de)dAs∣∣∣∣ Ft]

= eLgTE[Lγ,θ,εT

∫ T0Bs|Us(e)|φs(de)dAs

∣∣∣ Ft]Lγ,θ,εt

≤ eLgTE

[(Lγ,θ,εT )2

(Lγ,θ,εt )2

∣∣∣∣∣ Ft]1/2

·

E

[(∫ T

0

∫E

Bs|U is(e)|φs(de)dAs

)2∣∣∣∣∣ Ft

]1/2

,

where we used the Cauchy-Schwarz inequality. Choose δ > 0 such that 2Lf + δ < β,which is possible thanks to the conditions on β. The last term in the right hand side isP-a.s. finite as we have again by Cauchy-Schwarz inequality(∫ T

0

∫E

e−δAs/2eδAs/2|BsUis(e)|φs(de)dAs

)2

≤∫ T

0

e−δAsdAs

∫ T

0

∫E

eδAsB2s |U i

s(e)|2φs(de)dAs

≤ 1

δ

∫ T

0

∫E

e(δ+2Lf )As|U is(e)|2φs(de)dAs

<1

δ

∫ T

0

∫E

eβAs|U is(e)|2φs(de)dAs

which is in L1(P). The other term in the right hand side is bounded by E[eηAT

∣∣ Ft]which is also P-a.s. finite, as stated in lemma 2.11. Then (2.17) holds and the theoremis proven.

2.3.1 A monotonic limit result

We establish a monotonic limit proposition, as the one introduced in Peng [69], forBSDEs of a very particular type. The generator is linear in U , and it is the kind ofequation we will need later on in chapter 6. Consider the following sequence of BSDE

Y nt = ξ +

∫ T

t

fsdAs +

∫ T

t

∫E


∫ T

t

gsds

−∫ T

t

∫E

Uns q(dsde)−

∫ T

t

Zns dWs +Kn

T −Knt (2.18)

Kn are known non-decreasing predictable processes starting from zero that are squareintegrable, that is E [(Kn

T )2] <∞. In the following, the hypothesis on ξ, f and g are the

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ones already listed in this section for the existence and uniqueness of the BSDE drivenby marked point process and Wiener motion. Since f and g do not depend on (Y, U, Z)we only need them to be progressive processes, that for an appropriate β satisfy theintegrability condition

E[∫ T

0

eβAsf 2s dAs +

∫ T

0

eβAsg2sds

]<∞.

ρ is a P ⊗E-measurable random field such that 0 ≤ ρs(e) ≤ L for some constant L. Inthis subsection we are not concerned on the conditions for which that equation (2.18)has a solution, so we just assume it has one in the L2,β spaces for β > (max|L −1|, 1)2.

Proposition 2.16: Assume that for β > (max|L − 1|, 1)2, Y nt Yt for some Yt ∈

L2,β(A) ∩ L2,β(W ) and

‖Un‖L2,β(p) + ‖Zn‖L2,β(W ) ≤M.

Assume Kn are non decreasing cadlag predictable with Kn0 = 0 and E [(Kn

T )2] < ∞.Then there exists U ∈ L2,β(p), Z ∈ L2,β(W ) and K predictable cadlag increasingprocess with E[K2

T ] <∞, such that

Yt = ξ +

∫ T

t

fsdAs +

∫ T

t

∫E

Us(e)(ρs(e)− 1)φs(de)dAs +

∫ T

t

gsds

−∫ T

t

∫E


t

ZsdWs +KT −Kt.

Proof. First notice that Y is also the L2,β([0, T ]) limit of the Y n. Thanks to the boundon Un and Zn, we can extract a weakly convergent subsequence (Unk , Znk) to some(U,Z). This implies that for every fixed stopping time τ ,

∫ τ

0

∫E

Uns (e)q(dsde)

w

∫ τ

0

∫E

Us(e)q(dsde)∫ τ

0

Zns dWs

w

∫ τ

0

ZsdWs

since the application that associates∫ τ

0

∫EVs(e)q(dsde) ∈ L2(Fτ ) to V ∈ L2,β(p) is

continuous, and the same for the application that associates∫ τ

0XsdWs ∈ L2(Fτ ) to

X ∈ L2,β(W ) . Notice also that the linear application that to V ∈ L2,β(p) associates

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Chapter 2. Mathematical tools∫ τ0

∫EVs(e)(ρs(e)− 1)φs(de)dAs ∈ L2(Fτ ) is continuous, indeed:

E[∣∣∣∣∫ τ

0

∫E

Vs(e) (ρs(e)− 1)φs(de)dAs

∣∣∣∣2]

≤ E

[(∫ τ

0

∫E

|Vs(e)||(ρs(e)− 1)|φs(de)dAs)2]

≤ L′2E

[(∫ T

0

∫E

|Vs(e)|φs(de)dAs)2]

≤ L′2E[∫ T

0

e−βAsdAs

∫ T

0

∫E

eβAs|Vs(e)|2φs(de)dAs]

≤ L′2

βE[∫ T

0

∫E

eβAs|Vs(e)|2φs(de)dAs],

where L′ = max|L− 1|, 1. If we define Kτ as

Kτ = Y0 − Yτ −∫ τ

0

fsdAs −∫ τ

0

gsds−∫ τ

0

∫E

Us(e)(ρs(e)− 1)φs(de)dAs

+

∫ τ

0

∫E

Us(e)q(dsde) +

∫ τ

0

ZsdWs,

since Knτ satisfies

Knτ = Y n

0 − Y nτ −

∫ τ

0

fsdAs −∫ τ

0

gsds−∫ t

0

∫E

Uns (e)(ρs(e)− 1)φs(de)dAs

+

∫ τ

0

∫E

Uns (e)q(dsde) +

∫ τ

0

Zns dWs

it follows that Knkτ

w Kτ , for all τ . We can deduce that Kτ inherits the following

properties from Knτ :

• K0 = 0 and E[K2T ] <∞.

• K is increasing. Indeed consider σ, τ stopping times such that σ ≤ τ . Then wehave Kn

σ ≤ Knτ and for all X ∈ L2(Ω,F ,P), Z > 0 we have

E[KnσX] ≤ E[Kn

τX]⇒ E[KσX] ≤ E[KτX], (2.19)

since they converge weakly. Suppose exists an A ∈ F with P(A) > 0 such thatKτ −Kσ < 0 on A. Then by choosing X = 1A we have 0 >

∫A

(Kτ −Kσ)dP,which is a contradiction with the fact that

∫A

(Kτ −Kσ)dP ≥ 0 thanks to (2.19).

• K is predictable. This is a direct consequence of the fact that Kn w K in L2(Ω×

[0, T ],F ⊗ B([0, T ]),P⊗ λ) and P ⊂ F ⊗ B([0, T ]) (see 2.17). To show this letB ∈ L2(Ω× [0, T ]), for all t we know E[BtK

nt ]→ E[BtKt]. Then∫ T

0

E [Bt(Knt −Kt)] dt→ 0,

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2.4. Optimal stopping and optimal switching

thanks to dominated convergence theorem. That is,Kn w K inL2(Ω×[0, T ],F⊗

B([0, T ]),P⊗ λ).

Thanks to Lemma 2.2 in Peng [69], we also know that both Y and K are cadlag.

A short simple lemma that was used above:

Lemma 2.17: Let fnw f in L2(O,O, µ) with µ complete. Assume fn ∈ L2(O,U , µ)

with U ⊂ O. Then f ∈ L2(O,U , µ).

Proof. Since the measure is complete, L2(O,U , µ) is closed in L2(O,O, µ) (if f isan accumulation point for L2(O,U , µ), then for fn → f , it is possible to extract asubsequence that converges a.e. to f and thus f is U-measurable). So,(

L2(O,U , µ)⊥)⊥

= L2(O,U , µ).

Let h ∈ L2(O,U , µ)⊥, then

〈f, h〉 = limn〈fn, h〉 = 0,

that is f ∈(L2(O,U , µ)⊥

)⊥= L2(O,U , µ). Also, fn

w f in L2(O,U , µ).

2.4 Optimal stopping and optimal switching

In this work we will represent the solutions of an optimal stopping problem (in chapter3) and later of an optimal switching problem (in chapter 6) through the use of reflectedbackward SDEs. We place ourselves again in the framework described in 2.2. In thissection a brief introduction to this this kind of problems will be given. The readerfamiliar with this subject can skip this section entirely.

Optimal stopping A literature reference for optimal stopping is the book Peskir andShiryaev [71] or Karatzas and Shreve [55]. More general frameworks are treated in ElKaroui [27] and Kobylanski and Quenez [60]. Remember we denote by Tt the set of allstopping times greater than t. In this problem, a controller tries to maximise the outputof a stochastic system by choosing the moment to stop the system. Running gains perunit of time ft and gt are given, along an early stopping reward ht for stopping at timet and a final reward ξ if the system is not stopped. The controller chooses a stoppingtime τ to stop the system, obtaining a reward∫ τ∧T

t

fsdAs +

∫ τ∧T

t

gsds+ h1τ<T + ξ1τ≥T.

Of course we should consider the expected value (conditional to the filtration at timet) of the reward, that we denote by

J(t, τ) = E[∫ τ∧T

t

fsdAs +

∫ τ∧T

t


∣∣∣∣ Ft] .One aims at maximizing J over all possible stopping times, obtaining what is calledthe value function v

v(t) = ess supτ∈Tt

E[∫ τ∧T

t

fsdAs +

∫ τ∧T

t

gsds+ h1τ<T + ξ1τ≥T

∣∣∣∣ Ft] .33

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The aim of the problem is thus to characterize v and (if it exists) the optimal stoppingtime τ ∗, that is the time attaining the ess sup.

There are various classical approaches to this problem. One of the earliest ones, thatallows to solve the problem in a general setting, is the use of Snell envelope theory,detailed in the next section. It is basically the study of the properties of the ess supabove and its relation with the reward process. The downside is that it does not providea way of calculating v. In the case of a “Markovian” setting, that is when we havea forward process and the rewards are functions of this process, it is possible to usethe theory of Hamilton Jacobi Bellman equations. The function v is characterized assolution of a variational inequality. This approach cannot be used in our non-Markovianframework, so we do not detail it here. A comprehensive description can be found inBensoussan and Lions [6].

A third approach, the one adopted here for our framework and described in chapter3, is the use of reflected backward SDE. First studied in El Karoui et al. [30], in thiscase the function v is characterized as the solution of a backward stochastic equationwith a reflection on the barrier, which is the early stopping reward process. The optimalstopping time is then characterized thanks to the interaction between the solution of theRBSDE and the barrier.

Optimal switching Optimal switching can be thought as a generalization of optimalstopping, or as a particular case of impulse control. A stochastic system can evolvein m different modes, and to each mode it may correspond a different dynamic andgain functions. A controller chooses at which (random) times to switch and to whatmode, thus the name of the problem. We will leave the description of the particularswitching problem we face to the relevant chapter 6, and here we give a short andgeneral introduction where only the gains depend on the current mode.

The ingredients are:

• For each mode i ∈ 1, . . . ,m, gains f it and git.

• For each mode i ∈ 1, . . . ,m, a terminal gain ξi.

• For each couple of modes (i, j) ∈ 1, . . . ,m2, the cost Ct(i, j) of switching frommode i to mode j at time t.

As already anticipated, a controller chooses a sequence a = (τk, αk)k≥1 where τk arestopping times and αk are random variables that take values in 1, . . . ,m and indicateto what mode the system switches. We define a process indicating what strategy iscurrently active by

as =∑k≥1

αk−11(θk−1,θk](s)

Usually one defines a set Ait of admissible strategies starting at time t on mode i thatdepend on the specific problem. This may depend on some integrability condition onthe costs or other data of the particular problem. The controller seeks to maximize thequantity

J(t, i, a) = E

[ξaT +

∫ T

t

fass dAs +

∫ T

t

gass ds−∑k≥1

Cτk(αk−1, αk)

∣∣∣∣∣ Ft]

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2.5. Some remarks on the Snell envelope theory

over all possible strategies in Ait, obtaining the value function


J(t, i, a).

The objective of optimal switching is thus to characterize the value function v andpossibly find the optimal strategy a∗.

The dynamic of the underlying process may also change when changing swithcingmode. The classical example is when one has m possible forward Ito diffusions X i, sowe basically obtain a switched diffusionXa for each strategy. In this case the gains (andoften switching costs) are functions of this forward diffusion. In our case in chapter 6we will adopt a different idea, more suitable to marked point process: to each modecorresponds a different compensator and thus a different law.

This kind of problem has attracted a lot of attention. One of the first descriptions canbe found in Brennan and Schwartz [11]. Other cases in which optimal switching hasbeen treated are Brekke and Øksendal [9], Carmona and Ludkovski [12], Ludkovski[64], and Tang and Yong [79] among others.

The classical solution to solve the Markovian case with a forward diffusion is againthe use of a system of variational inequalities of the Hamilton Jacobi Bellman type, asdescribed in Bensoussan and Lions [7] for the more general case of impulse control.The viscosity solution to said equations represents the value function v and under suit-able conditions it is possible to find the optimal strategy too. A different technique, thatcan be found for example in Hamadene and Jeanblanc [49] or Djehiche, Hamadene,and Popier [26], is the use of a set of Snell envelope relations: m processes Ri aredefined where each one is the Snell envelope of the max over the other Rj . Again thishas the problem of not being completely operative. Another approach, followed by ushere, is the use of a system of reflected BSDE where in the obstacle of each componentof the system appears the best value among the solutions to the other equations of thesystem. Further and more precise details will be introduced in chapter 6.

2.5 Some remarks on the Snell envelope theory

The Snell envelope is a technique that was first developed in order to solve optimalstopping problems. The basic definition is that the Snell envelope R of a process X isthe smallest supermartingale that dominates said process, i.e. Rt ≥ Xt. Its general the-ory has been studied in various works. El Karoui [27] considers the case for a positiveprocess without any restrictions on the filtration, obtaining general results. For a bitless general results, but still enough for our work, Karatzas and Shreve [55] developsthe theory for non-negative cadlag processes, while Peskir and Shiryaev [71] treats thecase where the process is cadlag and left continuous over stopping times, and satisfiesthe condition

E[supt|ηt|]<∞. (2.20)

The recent work Kobylanski and Quenez [60] treats the subject in the framework offamily of random variables indexed by stopping times, using quite general assumptions.In the following, let (Ω,F ,P) be a probability space and let F = (Ft)t≥0 be a filtrationsatisfying the usual conditions. Let η be a cadlag process. Several properties that

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hold for positive processes can be shown under the condition (2.20), as we will see inproposition 2.19. We recall the following definition:Definition 2.18. An optional process R of class [D] is said to be regular if Rt− = pRt

for any t < T , where pX indicates the predictable projection.

Proposition 2.19: Let η be a cadlag process satisfying (2.20). Define

Rt = ess supτ∈Tt

E [ητ | Ft] (2.21)

It holds that

i) Rt is the Snell envelope of ηt. This means it is the smallest cadlag supermartingalethat dominates ηt, i.e. Rt ≥ ηt for all t P-a.s.

ii) A stopping time τ ∗ is optimal in (2.21) (i.e. Rt = E [ητ∗| Ft]) if and only if oneof the following conditions hold

• Rτ∗ = ητ∗ and Rs∧τ∗is a F-martingale• E[Rt] = E[η∗τ ]

iii) Rt is of class [D], hence it admits decomposition

Rt = Mt −Kt,

where M is a martingale, K a predictable increasing process with K0 = 0. K canbe decomposed as K = Kc

t +Kdt , where Kc indicates the continuous part and Kd

the discontinuous part. Moreover we have, a.s.

t : ∆Kt > 0 ⊂ t : Rt− = ηt−or equivalently, ∆Kt = ∆Kt1R(η)t−=ηt−, t ≥ 0.

iv) If the process Rt is regular in the sense that Rt− = pRt, where pR indicates thepredictable projection, defining the stopping time

D∗t = infs ≥ t : Rs 6= Ms,

then D∗t is an optimal stopping time and it is in fact the largest optimal stoppingtime.

Proof. DefineI = inf

t∈[0,T ]ηt and Nt = E [I| Ft] ,

and since ηt − I ≥ 0 for all t, we have ηt − Nt ≥ 0 for all t. Nt is a uniformlyintegrable martingale thanks to (2.20). Consider ηt = ηt −Nt ≥ 0 and Rt = Rt −Nt.Notice that then

Rt = Rt −Nt = ess supτ∈Tt

E [ητ −Nτ | Ft] = ess supτ∈Tt

E [ητ | Ft] ,

i.e. R is the Snell envelope of the positive process η. R inherits all the properties fromR. Let us see why the fourth property holds, as the rest are obtained similarly. If Rt

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2.5. Some remarks on the Snell envelope theory

is regular, so is Rt because we are adding a uniformly integrable martingale, which isregular (all uniformly quasi-left-continuous integrable cadlag martingales are regular,see He, Wang, and Yan [52] Def 5.49). The result then holds by El Karoui [27] pag140.

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CHAPTER3Reflected BSDE driven by marked point process

and optimal stopping

In this chapter we study the existence and uniqueness of a class of Reflected backwardSDE driven by a marked point process and a Brownian motion. This equation can thebe used to solve an optimal stopping problem which we will detail later.

In connection with optimal stopping and obstacle problems, in El Karoui et al. [30]a reflected BSDE is introduced, where the solution is forced to stay above a certaincontinuous barrier process. This class of BSDE finds applications in various problemsin finance and stochastic games theory. A number of generalizations has followed,both with variations on the nature of the barrier process and the type of noise. In theBrownian case, in Hamadene [44] the author solves the problem when the obstacle isjust cadlag. while in Peng and Xu [70] the authors allow the obstacle to be only L2. Onthe other hand, in Hamadene and Ouknine [46] the authors solve the problem when aPoisson noise is added, and the barrier is cadlag with inaccessible jump times. This islater generalized in Hamadene and Ouknine [47] where the barrier can have partiallyaccessible jumps too. Other specific results are Essaky and Hassani [37] where a BSDEwith two generators is solved in a Wiener framework and Ren and El Otmani [75] in aLevy framework; the papers Ren and Hu [76] and El Otmani [31] where the noise is aTeugels Martingale associated to a one-dimensional Levy process. The paper Crepeyand Matoussi [22] that considers a marked point process with compensator admitting abounded desnity with respect to the Lebesgue measure.

Finally, very general barriers beyond the cadlag case were recently considered inGrigorova et al. [42, 43].

It is the aim of the present work to address the case when the obstacle to be a cadlagprocess and, in addition to the Wiener process, a very general marked point process oc-curs in the equation. The only assumptions we make is that it is non-explosive and has

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Chapter 3. RBSDE driven by MPP and optimal stopping

totally inaccessible jumps. This is equivalent to the requirement that the compensatorof the counting process of the jumps has continuous trajectories. However, we do notrequire absolute continuiuty with respect to the Lebegue measure. To our knowledge,only in Bandini [1, 2], in Papapantoleon, Possamaı, and Saplaouras [65], and in Cohen,Elliott, and Pearce [17] and Cohen and Elliott [15] even more general cases have beenaddressed, but without reflection.

The equation has the form

Yt = ξ +

∫ T

t

fs(Ys, Us)dAs +

∫ T

t

gs(Ys, Zs)ds

−∫ T

t

∫E

Us(e)q(dtde)−∫ T

t

ZsdWs +KT −Kt

Yt ≥ ht.

(3.1)

Here W is a Brownian motion and q, independent from W , is a compensated inte-ger random measure corresponding to some marked point process (Tn, ξn)n≥1: seeBremaud [10], Jacod [54], and Last and Brandt [63] as general references on the sub-ject. The data are the final condition ξ and the generators f and g. A is a continuousstochastic increasing process related to the point process. The Y part of the solution isconstrained to stay above a given barrier process h, and the K term is there to assurethis condition holds. This equation is then used to solve a non-markovian optimal stop-ping problem, where the running gain, stopping reward and final reward are the dataused in the BSDE. Under additional assumptions on the barrier process, an optimalstopping time is characterized.

This work generalizes the results previously obtained by allowing a more generalstructure in the jump component. This introduces some technical difficulties and someassumptions. For instance, we work in “weighted L2 spaces”, with a weight of the formeβAt , and the data must satisfy this integrability conditions. Direct use of standard tools,like the Gronwall lemma, becomes difficult in our case, so we have to resort to directestimates. Since there is no general comparison theorem for BSDE with so generalmarked point process, we do not use a penalization method, but rather a combinationof the Snell envelope theory and contraction theorem.

The chapter is organized as follows: in section 3.1 we first describe the setting andthe problem we want to solve. In section 3.2 we prove the existence and uniquenessof a Reflected BSDE driven by a marked point process and a Wiener process when thegenerators do not depend on the solution of the BSDE. This is solved, as every timethere is Brownian motion involved, in the L2 spaces introduced in section 2.2. Whenthe (given) generator and the other data are adapted only to the filtration generatedby the point process, the solution can be found in a larger space. We then link theseequations to an optimal stopping problem. Lastly in section 3.3 we solve the BSDEin the general case with the help of a contraction argument. Here we need to use theL2 framework for both the case with only marked point process or with both drivingprocesses.

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3.1. Preliminaries, assumptions, formulation of the problems

3.1 Preliminaries, assumptions, formulation of the problems

We will place ourselves in the framework described in section 2.2. We will considerthe following reflected BSDE.

Yt = ξ +



∫ Tt

∫EUs(y)q(dsdy)


Yt ≥ ht, ∀t ∈ [0, T ] a.s.∫ T0


1Yt−=ht−∀t ∈ [0, T ] a.s.,

(3.2)

In section 2.2, we introduced the L2,β spaces. A solution is a quadruple (Y, U, Z,K)that lies in

(L2,β(A) ∩ L2,β(W )

)×L2,β(p)×L2β(W )×I2, with Y cadlag, that satisfies

(3.2). The condition on the last line in (3.2) is called the Skorohod condition, or theminimal push condition. It can be expressed in an alternative way: see Remark 3.2below.

Let us now state the general assumptions that will be used throughout the chapter.The first one is an assumption on the compensator A of the counting process N relativeto p, the second is about the data for the BSDE.

Assumption (3-A): The process A is continuous.

Assumption (3-B):

(i) The final condition ξ : Ω→ R is FT -measurable and

E[eβAT ξ2

]<∞.

(ii) For every ω ∈ Ω, t ∈ [0, T ], r ∈ R a mapping

f(ω, t, r, ·) : L2(E, E , φt(ω, dy))→ R

is given and satisfies the following:

a) for every U ∈ L2,β(p) the mapping

(ω, t, r) 7→ f(ω, t, r, Ut(ω, ·))

is Prog ⊗ B(R)-measurable, where Prog denotes the progressive σ-algebra.b) There exist Lf ≥ 0, LU ≥ 0 such that for every ω ∈ Ω, t ∈ [0, T ], y, y′ ∈ R,u, u′ ∈ L2(E, E , φt(ω, dy)) we have

|f(ω, t, y, u(·))− f(ω, t, y′, u′(·))| ≤

Lf |y − y′|+ LU

(∫E

|u(e)− u′(e)|2φt(ω, de))1/2

c) we have

E[∫ T

0

eβAs|f(s, 0, 0)|2dAs]<∞.

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(iii) The mapping g : Ω× [0, T ]× R× Rd → R is given

a) g is Prog × B(R)× B(Rd)-measurable.

b) There exist Lg ≥ 0, LZ ≥ 0 such that for every ω ∈ Ω, t ∈ [0, T ], y, y′ ∈ R,z, z′ ∈ Rd

|g(ω, t, y, z)− g(ω, t, y′, z′)| ≤ Lg|y − y′|+ LZ |z − z′|

c) we have

E[∫ T

0

eβAs|g(s, 0, 0)|2ds]<∞.

(iv) h is a cadlag F-adapted process such that hT ≤ ξ. There exists a δ > 0 such that

E[ supt∈[0,t]

e(β+δ)At |ht|2]

Remark 3.1. We recall that Assumption (3-A) is equivalent to the fact that the jumpsof the point process are totally inaccessible (relative to F): see He, Wang, and Yan [52]Corollary 5.28. We will often use the following consequence: since K is required to bepredictable, its jumps (that are all non-negative) are disjoint from the jumps of p; so atany jump time of K we also have a jump of Y with the same size, but of opposite sign,in symbols we have a.s.

∆Kt1∆Kt>0 = (−∆Yt)+1∆Kt>0, t > 0. (3.3)

Remark 3.2. The Skorohod condition on the last line in (3.2) tells us that the processK grows only when the solution is about to touch the barrier. We claim that it is in factequivalent to ∫ T

0

(Ys− − hs−)dKs = 0, a.s. (3.4)

To check the equivalence, note first that∫ T

0

(Ys− − hs−)dKs =

∫ T

0

(Ys − hs)dKcs +

∑0<s≤T

(Ys− − hs−)∆Ks, a.s.

If the Skorohod condition in (3.2) holds then both terms in the right-hand side are zero,since jumps of K can only happen when Yt− = ht− . Conversely, assume that (3.4)holds. Then clearly

∫ T0

(Ys− − hs−)dKcs = 0 and so

∫ T0

(Ys − hs)dKcs = 0. Also,∑

0<s≤T (Ys− − hs−)∆Ks = 0, so t : ∆Kt > 0 ⊂ t : Yt− = ht− and, recalling(3.3), we have a.s.

∆Kt = ∆Kt1∆Kt>0 = (−∆Yt)+1∆Kt>0 ≤ (−∆Yt)

+1Yt−=ht−

= (Yt− − Yt)+1Yt−=ht− = (ht− − Yt)+

1Yt−=ht−.

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3.2. Reflected BSDE with known generators and optimal stopping problem

Remark 3.3. In the simpler case when there is no Brownian component the reflectedBSDE (3.2) becomesYt = ξ +

∫ Ttf(s, Ys, Us)dAs −

∫ Tt

∫EUs(y)q(dsdy) +KT −Kt, ∀t ∈ [0, T ] a.s.

Y cadlag and Y ∈ L2,β(A,G), U ∈ L2,β(p,G), K ∈ I2(G)

Yt ≥ ht ∀t ∈ [0, T ] a.s.∫ T0


1Yt−=ht−∀t ∈ [0, T ] a.s.(3.5)

Here we only assume we are given the space (Ω,F ,P) and the marked point process p.The assumptions we need are the same as in (3-A) and (3-B), provided we set g = 0and F = G.

3.2 Reflected BSDE with known generators and optimal stopping prob-lem

In this section we first study the reflected BSDE in the case when the generators g andf do not depend on (Y, Z, U) but are a given processes that satisfy

Assumption (3-B′): f and g are F-progressive processes such that for some β > 0

E[∫ T

0

eβAs |fs|2dAs +

∫ T

0

eβAs|gs|2ds]<∞. (3.6)

Equation (3.2) reduces toYt = ξ +

∫ TtfsdAs +

∫ Ttgsds−

∫ Tt

∫EUs(y)q(dsdy)−


Y ∈ L2,β(A) ∩ L2,β(W ), U ∈ L2,β(p), Z ∈ L2,β(W ), K ∈ I2

Yt ≥ ht ∀t ∈ [0, T ] a.s.∫ T0


1Yt−=ht−∀t ∈ [0, T ] a.s.(3.7)

In this case, the solution Y to the equation is also the value function of an optimalstopping problem, as we will see later. First we define the cadlag process ηt as

ηt =

∫ t∧T

0

fsdAs +

∫ t∧T

0

gsds+ ht1t<T + ξ1t≥T (3.8)

Remark 3.4. In the following we will often use this kind of inequalities:(∫ t

0

fsdAs

)2

=

(∫ t

0

e−βAs/2eβAs/2|fs|dAs)2

≤∫ t

0

e−βAsdAs

∫ t

0

eβAsf 2s dAs

=1− eβAt

β

∫ t

0

eβAsf 2s dAs ≤

1

β

∫ t

0

eβAsf 2s dAs (3.9)

Lemma 3.5: Under assumptions (3-B)-(i)(iv) and (3-B′), η is of class [D] and

E[

sup0≤t≤T

|ηt|2]<∞

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Proof. Fix a stopping time τ . Clearly

|ητ |2 ≤ 4

(∫ T

0

|fs|dAs)2

+ 4

(∫ T

0

|gs|ds)2

+ 4|hτ |21τ<T + 4|ξ|2

≤ 4

β

∫ T

0

eβAsf 2s dAs + 4T

∫ T

0

eβAs|gs|2ds+ 4 supt∈[0,T ]

eβAt |ht|2 + 4eβAT ξ2,

(3.10)

and since the right-hand side has finite expectation we obtain the class [D] property.Likewise, by taking the supremum over all t ∈ [0, T ], and expectation after that, weobtain the second property.

Now, using the Snell envelope theory, we show that there exists a solution to theequation above. Section 2.5 lists the properties that we will need in the following.

Proposition 3.6: Let assumptions (3-A), (3-B)-(i)(iv) and (3-B′) hold for some β > 0,then there exists a unique solution to (3.7).

Proof. The uniqueness property is stated and proved separately in Proposition 3.9 be-low. Existence is proved in several steps.Step 1. We start by defining Yt, for all t ≥ 0, as the optimal value of the stoppingproblem:

Yt = ess supτ∈Tt

E[∫ τ∧T

t

fsdAs +

∫ τ∧T

t


∣∣∣∣ Ft] . (3.11)

From (3.10) it follows that Yt is integrable for all t and Yt = ξ for t ≥ T . We have thefollowing a priori estimate on Y , that we will prove later.

Lemma 3.7: Assume (3-B)-(i)(iv) and (3-B′) above on ξ, f, h, ξ. Then

E

[supt∈[0,T ]

eβAtY 2t

]<∞. (3.12)

It follows that

Yt +

∫ t∧T

0

fsdAs +

∫ t∧T

0

gsds = ess supτ∈Tt

E [ητ | Ft]

so Yt +∫ t∧T

0fsdAs +

∫ t∧T0

gsds is the Snell envelope of η, that is the smallest super-martingale such that Yt +

∫ t∧T0

fsdAs +∫ t∧T

0gsds ≥ ηt. Since η is cadlag, its Snell

envelope R(η), and hence Y , have a cadlag modification. We refer to section 2.5 for areview of the properties of the Snell envelope that we will use. Also, from now on allsupermartingales that we consider in this proof are assumed to be cadlag. Also, since ηsatisfies (2.20) by Lemma 3.5, Y +

∫ ·∧T0

fsdAs +∫ ·∧T

0gsds is of class [D] and thus it

admits a unique Doob-Meyer decomposition

Yt +

∫ t∧T

0

fsdAs

∫ t∧T

0

gsds = Mt −Kt, (3.13)

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where M is a martingale and K is a predictable increasing process starting from zero.From Lemma 3.5 and the result in Dellacherie and Meyer [25, p 220, Chapter VII] itfollows that EK2

T < ∞, so that M is a square integrable martingale. Furthermore,K can be decomposed into Kc + Kd, the continuous and discontinuous part, and wehave that ∆Kt = ∆Kt1R(η)t−=ηt− (see 2.19.iii)). However it is immediate to see thatR(η)t− = ηt− if and only if Yt− = ht−1t≤T + ξ1t>T and it follows that

∆Kt = ∆Kt1Yt−=ht−, t ∈ [0, T ]. (3.14)

By the martingale representation theorem, there exists some U and Z such that

E[∫ T

0

∫E

|Ut(e)|φt(de)dAt]

+ E[∫ T

0

|Zt|2dt]<∞ (3.15)

Mt = M0 +

∫ t

0

∫E

Us(e)q(dsde) +

∫ t

0

ZsdWs. (3.16)

Choosing τ = t in (3.11) we see that a.s. Yt ≥ ht for all t < T and YT = ξ, so Yt ≥ htfor all t ≤ T a.s. Plugging (3.15) in (3.13) we conclude that the first equality in (3.7) isverified.

Step 2. In this step we prove that the Skorohod conditions in (3.7) hold. From (3.3)it follows that ∆Kt ≤ (−∆Yt)

+ and, taking into account (3.14), we obtain

∆Kt ≤ (−∆Yt)+1Yt−=ht− = (Yt− − Yt)+

1Yt−=ht−, (3.17)

that gives us the second condition. Consider now Yt = Yt +∫ t

0fsdAs +

∫ t0gsds +

Kdt = Mt − Kc

t and ηt = ηt + Kdt . We claim that Yt is the Snell envelope of ηt.

Indeed, it is a supermartingale that dominates ηt. Let Qt be another supermartingalethat dominates ηt. Then Qt − Kd

t is still a supermartingale, and dominates ηt. Then,since Yt+

∫ t0fsdAs+

∫ t0gsds = R(η)t,Qt ≥ Yt. Then Yt is the smallest supermartingale

that dominates ηt, and thus its Snell envelope. Next, Yt +∫ t

0fsdAs + Kd

t = Mt −Kct

is regular (we recall that a process X is regular if Xt− = pXt, where pXt denotes thepredictable projection, see also 2.19.iv); all uniformly integrable cadlag martingales areregular). Then, the stopping time defined as

D∗t = inf s ≥ t : Ms 6= R(η)s = inf s ≥ t : Kcs > Kc

t

is the largest optimal stopping time, and it satisfies:

YD∗t = ηD∗t

Ys∧D∗t is a F-martingale

See (2.19.ii)). Define then

Dt = infs ≥ t : Ys ≤ ηs

Since YD∗t = ηD∗t we have Dt ≤ D∗t , and it follows that

0 =

∫ Dt

t

(Ys − ηs

)dKc

s =

∫ Dt

t

(Ys − hs) dKcs ,

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which impliesKcDt

= Kct for arbitrary t, and hence

∫ T0

(Ys − hs) dKcs = 0, that together

with (3.17) gives us the Skorohod conditions.

Step 3. We conclude the proof showing that the processes are in the right spaces.We have already noticed that E[K2

T ] < ∞. Next we define the sequence of stoppingtimes:

Sn = inf

t ∈ [0, T ] :

∫ t

0

eβAs |Ys|2dAs +

∫ t

0

∫E


+

∫ t

0

eβAs |Zs|2ds > n

,

and consider the “Ito Formula” applied to eβ(At+t)Y 2t between 0 and Sn. We have

eβ(ASn+Sn)Y 2Sn = Y 2

0 + β

∫ Sn

0


∫ Sn

0

eβ(As+s)Y 2s ds

+ 2

∫ Sn

0

∫E


∫ Sn

0

eβ(As+s)YsZsdWs

− 2

∫ Sn

0


∫ Sn

0

eβ(As+s)Ysgsds

− 2

∫ Sn

0

eβ(As+s)Ys−dKs +

∫ Sn

0

eβ(As+s)Z2sds

+∑

0<s≤Sn

eβ(As+s)∆K2s +

∫ Sn

0

∫E

eβ(As+s)U2s (e)p(dsde)

Now we use the fact that

∫ t

0

∫E

Us(e)p(dsde) =

∫ t

0

∫E

Us(e)φs(de)dAs +

∫ t

0

∫E

Us(e)q(dsde),

and, by Remark 3.2,

∫ t

0

eβ(As+s)Ys−dKs =

∫ t

0

eβ(As+s)(Ys− − hs−)dKs

=0

+

∫ t

0

eβ(As+s)hs−dKs.

Neglecting the positive terms Y 20 and

∑0<s≤Sn e

β(As+s)∆K2s the previous equation be-

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comes

eβ(ASn+Sn)Y 2Sn ≥ β

∫ Sn

0


∫ Sn

0

eβ(As+s)Y 2s ds

+ 2

∫ Sn

0


∫ Sn

0

eβ(As+s)YsZsdWs

− 2

∫ Sn

0


∫ Sn

0

eβ(As+s)Ysgsds

− 2

∫ Sn

0

eβ(As+s)hs−dKs +

∫ Sn

0

∫E


+

∫ Sn

0

∫E

eβ(As+s)U2s (e)q(dsde) +

∫ Sn

0

eβ(As+s)Z2sds,

By the definition of Sn and remembering that Y satisfies (3.12), and using Burkholder-Davis-Gundy inequality we have that

∫ t∧Sn

0

eβ(As+s)YsZsdWs

is a martingale. Indeed we have

E

[supt∈[0,T ]

∣∣∣∣∫ t∧Sn

0

eβ(As+s)YsZsdWs

∣∣∣∣]≤ E

[(∫ Sn

0

e2β(As+s)Y 2s Z

2sds

)1/2]

≤ eβTE

[supteβAt/2|Yt|

(∫ Sn

0

eβAsZ2sds

)1/2]

≤ n1/2eβTE[supteβAtY 2

t

]<∞. (3.18)

Similarly, since

E[∫ t

0

∫E

eβ(As+s)|Ys−Us(e)|φs(de)dAs]≤ E

[∫ t

0

eβ(As+s)Y 2s dAs

]+ E

[∫ t

0

∫E


]≤ 2n <∞, (3.19)

we obtain that∫ t∧Sn

0

∫Eeβ(As+s)Ys−Us(e)q(dsde) is a martingale. Reordering terms

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and taking expectation we obtain

βE[∫ Sn

0

eβ(As+s)Y 2s dAs

]+ E

[∫ Sn

0

∫E


]+ βE

[∫ Sn

0

eβ(As+s)Y 2s ds

]+ E

[∫ Sn

0

eβ(As+s)Z2sds

]≤ E

[eβ(ASn+Sn)Y 2

Sn

]+ 2E

[∫ Sn

0

eβ(As+s)YsfsdAs

]+ 2E

[∫ Sn

0

eβ(As+s)Ysgsds

]+ 2E

[∫ Sn

0

eβ(As+s)hs−dKs

]≤ E

[supteβ(At+t)Y 2

t

]+β

2E[∫ Sn

0

eβ(As+s)Y 2s dAs

]+β

2E[∫ Sn

0

eβ(As+s)Y 2s ds

]+

1

βE[∫ T

0

eβ(As+s)f 2s dAs

]+

2

βE[∫ T

0

eβ(As+s)g2sds

]+ γE

[supte(β+δ)(At+t)h2

t−

]+

1

γE

[(∫ Sn

0

e(β−δ)As+s2 dKs

)2],

(3.20)

where γ > 0 is a constant whose value will be chosen sufficiently large afterwards. Weonly need to estimate the last term with the integral in dK. In order to do that we applyIto’s formula to e(β−δ)At+t

2 Yt between 0 and a stopping time τ , obtaining the followingrelation(∫ τ

0

e(β−δ)As+s2 dKs

)2

=

(Y0 − e(β−δ)Aτ+τ

2 Yτ +β − δ

2

∫ τ

0

e(β−δ)As+s2 YsdAs

+β − δ

2

∫ τ

0

e(β−δ)As+s2 Ysds−

∫ τ

0

e(β−δ)As+s2 fsdAs

−∫ τ

0

e(β−δ)As+s2 gsds+

∫ τ

0

∫E

e(β−δ)As+s2 Us(e)q(dsde)

+

∫ τ

0

e(β−δ)As+s2 ZsdWs

)2

Notice that the following holds:(∫ τ

0

e(β−δ)As+s2 YsdAs

)2

≤∫ τ

0

e−δ(As+s)dAs

∫ τ

0

eβ(As+s)Y 2s dAs

≤ 1

δ

∫ τ

0

eβ(As+s)Y 2s dAs(∫ τ

0

e(β−δ)As+s2 Ysds

)2

≤∫ τ

0

e−δAse−δsds

∫ τ

0

eβ(As+s)Y 2s ds

≤ 1

δ

∫ τ

0

eβ(As+s)Y 2s ds

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and similarly (∫ τ

0

e(β−δ)As+s2 fsdAs

)2

≤ 1

δ

∫ τ

0

eβ(As+s)f 2s dAs(∫ τ

0

e(β−δ)As+s2 gsds

)2

≤ 1

δ

∫ τ

0

eβ(As+s)g2sds

We note that for a P ⊗ E measurable process H we have

E

[(∫ t

0

∫E

Hs(e)q(dsde)

)2]≤ E

[∫ t

0

∫E

H2s (e)φs(de)dAs

],

as seen in proposition 2.6. Taking expectation and using Ito isometry we obtain the

bound for(∫ τ

0e(β−δ)As+s

2 dKs

)2

:

E

[(∫ τ

0

e(β−δ)As+s2 dKs

)2]≤ 16E

[supteβ(At+t)Y 2

t

]+

8

δE[∫ τ

0

eβ(As+s)g2ss

]+ 2

(β − δ)2

δE[∫ τ

0

eβ(As+s)Y 2s ds

]+ 2

(β − δ)2

δE[∫ τ

0

eβ(As+s)Y 2s dAs

]+

8

δE[∫ τ

0

eβ(As+s)f 2s dAs

]+ 8E

[∫ τ

0

eβ(As+s)Z2sds

]+ 8E

[∫ τ

0

∫E


].

By plugging this last estimate into (3.20), by choosing α, γ such that

γ > max

(8, 4

(β − δ)2

βγ

)we obtain

E[∫ Sn

0

eβ(As+s)Y 2s dAs

]+ E

[∫ Sn

0

eβ(As+s)Y 2s ds

]+ E

[∫ Sn

0

∫E

eβ(As)U2s (e)φs(de)dAs

]+ E

[∫ Sn

0

eβ(As)Z2sds

]≤ C

(E[supteβAtY 2

t

]+ 2

(1

β+

1

δγ

)E[∫ T

0

eβAsf 2s dAs

]+E

[∫ T

0

eβAsg2sds

]+ γE

[supte(β+δ)Ath2

t−

]),

for some constant C independent of n. Now let S = limn Sn and by the last estimate,considering how Sn are defined, we have S = T . This implies that Y ∈ L2,β(A) ∩L2,β(W ), Z ∈ L2,β(W ) and U ∈ L2,β(p).

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Proof of lemma 3.7. By the definition of Y we have

eβAt/2|Yt| ≤ E[eβAT /2|ξ|+ eβAt/2

∫ T

t

|fs|dAs

+ eβAt/2∫ T

t

|gs|ds+ sup0≤s≤T

eβAs/2|hs|∣∣∣∣Ft]

Proceeding as in Remark 3.4 we have∫ T

t

|fs|dAs ≤e−βAt/2

β1/2

(∫ T

t

eβAs |fs|2dAs)1/2

and it follows that

eβAt/2|Yt| ≤ E

[eβAT /2|ξ|+ 1

β1/2

(∫ T

0

eβAs|fs|2dAs)1/2

+

∫ T

0

eβAs/2|gs|ds+ sup0≤s≤T

eβAs/2|hs|∣∣∣∣Ft] =: St

Under assumption (3-B)-(i)(iv) and (3-B′), S is a square integrable martingale. Then

by Doob’s martingale inequality E[

sup0≤t≤T

eβAt |Yt|2]≤ CE [S2

T ] <∞.

Remark 3.8. Contrary to the diffusive (or diffusive and Poisson) case, the fact thatE[supt∈[0,T ] e

βAtY 2t

]< ∞ does not imply that Y ∈ L2,β(A). For this to happen we

would need additional conditions on A, for example E[A2T ] <∞.

Next we prove uniqueness.

Proposition 3.9: Let assumptions (3-A), (3-B)-(i)(iv) and (3-B′) hold for some β > 0,then the solution to (3.7) is unique.

Proof. Let (Y ′, U ′, Z ′, K ′) and (Y ′′, U ′′, Z ′′, K ′′) be two solutions. Define

Y = Y ′ − Y ′′ U = U ′ − U ′′ Z = Z ′ − Z ′′ K = K ′ −K ′′,

then (Y , U , Z, K) satisfies

Yt = −∫ T

t

∫E

U(e)q(dsde)−∫ T

t

ZsdWs + KT − Kt. (3.21)

We compute d(eβ(At+t)Y 2t ) by the Ito formula and we obtain

− Y 20 = β

∫ T

0


∫ T

0

eβ(As+s)Y 2s ds− 2

∫ T

0

Ys−dKs

+ 2

∫ T

0

∫E

eβ(As+s)Ys−Us(y)q(dsdy) +

∫ T

0

eβ(As+s)YsZsdWs

+

∫ T

0

eβ(As+s)Z2sds+

∑0<s≤T

eβ(As+s)(∆Ys)2 (3.22)

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The last term can be divided in totally inaccessible jumps (from the martingale inq(dsde)) and predictable jumps, from the K process, thus:∑

0<s≤T

eβ(As+s)(∆Ys)2 ≥

∑0<Tn≤T

eβ(As+s)U2Tn(ξn) =

∫ T

0

∫E

U2s (e)p(dsde)

=

∫ T

0

∫E

U2s (e)q(dsde) +

∫ T

0

∫E

U2s (e)φs(de)dAs

Proceeding as in (3.18) and (3.19) we prove that the stochastic integrals with respect toW and q are martingales. By neglecting Y 2

0 and taking expectation in (3.22), we obtain

βE[∫ T

0

eβ(As+s)Y 2s dAs

]+ βE

[∫ T

0

eβ(As+s)Y 2s ds

]+ E

[∫ T

0

∫E

eβ(As+s)U2s (y)φs(dy)dAs

]+ E

[∫ T

0

eβ(As+s)Z2sds

]≤ 2E

[∫ T

0

eβ(As+s)Ys−dKs

].

Now, taking into account Remark 3.2 we have∫ T

0

Ys−dKs =

∫ T

0

(Y ′s− − hs−)dK ′s

=0

−∫ T

0

(Y ′s− − hs−)dK ′′s

≥0

+

−∫ T

0

(Y ′′s− − hs−)dK ′s

≥0

+

∫ T

0

(Y ′′s− − hs−)dK ′′s

=0

≤ 0,

and thus

β||Y ||2L2,β(A) + β||Y ||2L2,β(W ) + ||U ||2L2,β(p) + ||Z||2L2,β(W ) ≤ 0,

which gives the uniqueness of Y , U and Z. From (3.21) we obtain

KT = Kt ∀t ∈ [0, T ].

Then KT = 0 since K0 = 0 and consequently Kt = 0 for all t.

Consider now the optimal stopping problem with running gains f, g, early stoppingreward h and non stopping reward ξ. This means we are interested in the quantity

v(t) = ess supτ∈Tt

E[∫ τ

t

fsdAs +

∫ τ

0


∣∣∣∣ Ft] .Notice that we have two running gains, f integrated with respect to the process A, andg integrated with respect to Lebesgue measure in time. This could be used for exampleif we want to describe two different time dynamics, one depending on the speed of thepoint process.

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It is possible to show that the solution to the RBSDE solves the optimal stopping prob-lem and it is possible to identify an ε-optimal stopping time. Under additional assump-tions, it is possible to find an optimal stopping time. For this we need a definition,given in Kobylanski and Quenez [60] for admissible families over stopping times, thatwe adapt to our simpler case:

Definition 3.10. We say that a process φ is left (resp. right) upper semi-continuousover stopping times in expectation (USCE) if for all θ ∈ T0, E [φθ] < ∞ and for allsequences of stopping times (θn) such that θn ↑ θ (resp. θn ↓ θ) it holds that

E[φθ] ≥ lim supn→∞

E[φθn ].

Remark 3.11. If φ is a left upper semi continuous progressive process, then φ is leftupper semi continuous along stopping times. If also E[supt |φt|] holds, then it is leftUSCE. Indeed we have

lim supn→∞

E [φθn ] ≤ E[lim supn→∞

φθn

]≤ E [φθ] .

by using Reverse Fatou’s lemma with supt |φt| as dominant.

Proposition 3.12: Let assumptions (3-A), (3-B)-(i)(iv) and (3-B′) hold. Then we have:

1. The solution to the RBSDE (3.7) is a solution to the optimal stopping problem

Yt = ess supτ∈Tt

E[∫ τ

t

fsdAs +

∫ τ

0


∣∣∣∣ Ft] .2. For all ε > 0, define Dε

t as

Dεt = inf s ≥ t : Ys ≤ hs + ε ∧ T.

Then Dεt is an ε-optimal stopping time in the sense that

Yt ≤ ess supτ∈Tt

E[∫ Dεt

t

fsdAs +

∫ Dεt

0

gsds+ hDεt1Dεt<T + ξ1Dεt≥T

∣∣∣∣ Ft]+ ε.

3. If in addition ht1t<T + ξ1t≥T is left USCE, then

τ ∗t = inf s ≥ t : Ys ≤ hs ∧ T.

is optimal and is the smallest of all optimal stopping times.

Remark 3.13. The condition on the third point may seem unusual, but it is satisfied forexample if ht is left upper semi continuous on [0,T] and hT <≤ ξ.

Proof. Let τ ∈ Tt and consider the first equation (3.7) between t and τ :

Yt = Yτ +

∫ τ

t

fsdAs +

∫ τ

t

gsds−∫ τ

t

∫E

Zs(y)q(dsdy) +Kτ −Kt.

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By taking conditioning at Ft we have

Yt = E[Yτ +

∫ τ

t

fsdAs +

∫ τ

t

gsds+Kτ −Kt

∣∣∣∣ Ft] (3.23)

≥ E[hτ1τ<T + ξ1τ≥T +

∫ τ

t

fsdAs +

∫ τ

t

gsds

∣∣∣∣ Ft] , (3.24)

since the integral on q is a martingale, K is increasing and Yt ≥ hτ1t<T + ξ1t=T.To prove the reverse inequality, consider ε > 0 and the corresponding Dε

t . It holdsthat YDεt ≤ hDεt + ε on Dε

t < T. And on Dεt = T we have Yu > hu + ε for all

t ≤ u < T . Then , between t and Dεt , Ys− > hs− and thus∫ Dεt

t

(Ys− − hs−)dKs = 0⇒ KDεt= Kt.

Considering all this in (3.23) we have

Yt = E[YDεt +

∫ Dεt

t

fsdAs +

∫ Dεt

t

gsds

∣∣∣∣ Ft]≤ E

[hDεt1Dεt<T + ξ1Dεt=T +

∫ Dεt

t

fsdAs +

∫ Dεt

t

gsds

∣∣∣∣ Ft]+ ε. (3.25)

This together with (3.24) proves points one and two. For the third point, notice that Dεt

are non increasing in ε and that Dεt ≤ τ ∗. Thus Dε

t → D0t ≤ τ ∗ when ε → 0. Now

since ht1t<T + ξ1t=T is left USCE and the integral part is too, we have from (3.25)

E[Yt] ≤ lim supε→0

E[hDεt1Dεt<T + ξ1Dεt=T +

∫ Dεt

t

fsdAs +

∫ Dεt

t

gsds

]≤ E

[hD0

t1D0

t<T + ξ1D0t=T +

∫ D0t

t

fsdAs +

∫ D0t

t

gsds

].

Thus we have

E [Yt] = E

[hD0

t1D0

t<T + ξ1D0t=T +

∫ D0t

t

fsdAs +

∫ D0t

t

gsds

],

so D0t is optimal (see 2.19.ii)). We only need to prove that D0

t = τ ∗. We already knowthat D0

t ≤ τ ∗. On the other hand, since D0t is optimal it holds that YD0

t= ηD0

t, and

thus by the definition of τ ∗, τ ∗ ≤ D0t . This also proves that τ ∗ is the smallest optimal

stopping time.

A further interesting property holds when the reward is left USCE:

Proposition 3.14: Under assumptions (3-B)-(i)(iv) and (3-B′), if hτ1τ<T + ξ1τ≥Tis also left USCE, then K in the solution of (3.7) is continuous.

Proof. The proof is given in Kobylanski and Quenez [60] in the case were the rewardis a positive progressive process φ of class [D]. We can adapt to our case by using the

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transformation

I = inftηt Nt = E [I| Ft] ηt = ηt −Nt.

We have that ηt is USCE, as E[ηt] = E[ηt]− E[I]. Indeed let θn ↑ θ, then

lim supn→∞

E[ηθn ] ≤ E[ηθ].

Then if R(η) denotes the Snell envelope of η, it holds that R(η) = R(η) − Nt. TheDoob-Meyer decomposition for the cadlag supermartingale R(η) holds:

R(η)t = Mt − Kt

With K continuous thanks to Proposition B.10 in Kobylanski and Quenez [60]. ThenYt +

∫ t0fsdAs = R(η) = R(η) + Nt = Mt + Nt − Kt, but since the decomposition

is unique,∫ t

0

∫EZs(y)q(dsdy) = Mt = M + Nt and Kt = Kt. Thus the term K is

continuous.

If we are interested only in (3.5), and we have a filtration generated only by a MPPand g ≡ 0, the proofs above are still applicable. In this case, there is no particularreason to use a L2 space, since the martingale representation theorem for marked pointprocesses works in L1 (see Jacod [54]). We thus obtain the following:

Proposition 3.15: Let assumption (3-A) hold. Let ξ be a GT -measurable random vari-able. Let f, h be G-progressive processes. Assume that

E

[|ξ|+

∫ T

0

|fs|dAs + supt∈[0,T ]

|ht|

]<∞.

Then there exists a unique solution to the systemYt = ξ +

∫ TtfsdAs −

∫ Tt

∫EUs(y)q(dsdy) +KT −Kt

Yt ≥ ht ∀t ∈ [0, T ] a.s.∫ T0

(Ys − hs)dKcs = 0 and ∆Ks ≤ (hs− − Ys)+

1Ys−=hs−.

(3.26)

where Y is a cadlag G-adapted process such that E [|Yt|] < ∞ for all t, K is a G-predictable cadlag increasing process withK0 = 0 and E [KT ] <∞ and U is a P(G)⊗E-measurable process such that E

[∫ T0


]<∞.

Proof. Existence of a solution is obtained as in 3.6. The process ηt satisfies then theweaker condition E [supt |ηt|] < ∞, but this is enough to apply the Snell’s enveloperesults (see section 2.5, in particular (2.20)). Integrability is straightforward. Now let(Y ′, U ′, K ′) and (Y ′′, U ′′, K ′′) be two solutions, their difference satisfies

Y ′t − Y ′′t = Y ′0 − Y ′′0 +

∫ t

0

∫E

(U ′s(e)− U ′′s (e))q(dsde)− (K ′t −K ′′t ). (3.27)

Uniqueness of the component Y comes from the fact that if (Y, U,K) satisfies theequation, the cadlag process Y satisfies

Yt = ess supτ∈Tt

[∫ τ∧T

t

fsdAs + hτ1τ<T + ξ1τ≥T

∣∣∣∣Gt] ,54

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3.3. Reflected BSDE in the general case

which can be shown as in proposition 3.12, adapted to the this case with less integra-bility. Relation (3.27) becomes∫ t

0

∫E

U ′s(e)q(dsde)−K ′t =

∫ t

0

∫E

U ′′s (e)q(dsde)−K ′′t .

Since the predictable jumps of K and the totally inaccessible jumps of the integralswith respect to q are disjoint, we have that U ′Tn(ξn) = U ′′Tn(ξn) for all n. Then

E[∫ T

0

∫E

|U ′s(e)− U ′′s (e)|φs(de)dAs]

= E[∫ T

0

∫E

|U ′s(e)− U ′′s (e)|p(dsde)]

= E

[∑n≥1

|U ′Tn(ξn)− U ′′Tn(ξn)|

]= 0,

and thus U ′s(e) = U ′′s (e) φs(de)dAsdP-a.e. Then K ′t = K ′′t a.s. and uniqueness isproven.

We have then a result for optimal stopping analogous to proposition 3.12:

Proposition 3.16: Assume that the conditions of proposition 3.15 hold. Then

1. The solution to the RBSDE (3.26) is a solution to the optimal stopping problem

Yt = ess supτ∈Tt

E[∫ τ

t

fsdAs +

∫ τ

0


∣∣∣∣ Ft] .2. For all ε > 0, define Dε

t as

Dεt = inf s ≥ t : Ys ≤ hs + ε ∧ T.

Then Dεt is an ε-optimal stopping time in the sense that

Yt ≤ ess supτ∈Tt

E[∫ Dεt

t

fsdAs +

∫ Dεt

0

gsds+ hDεt1Dεt<T + ξ1Dεt≥T

∣∣∣∣ Ft]+ ε.

3. If in addition ht1t<T + ξ1t≥T is left USCE, then

τ ∗t = inf s ≥ t : Ys ≤ hs ∧ T.

is optimal and is the smallest of all optimal stopping times. Moreover, the processK is continuous.

3.3 Reflected BSDE in the general case

We now turn to the case where the generators depend on the solution, that is equa-tion (3.2). Denote by λ the Lebesgue measure on [0, T ], and introduce now L2,β(Ω ×[0, T ],F ⊗B([0, T ]), (A(ω, dt) + λ(dt)), the space of all F-progressive processes suchthat

‖Y ‖2L2,β(A+λ) = E

[∫ T

0

eβAsY 2s (dAs + ds)

]<∞.

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For brevity we denote is as L2,β(A+ λ) in the following. It is a Hilbert space equippedwith the norm above. It is clear that a process is in L2,β(A + λ) if and only if lies inY ∈ L2,β(A) ∩ L2,β(W ).

Theorem 3.17: Let assumption (3-A) and (3-B) hold for some β > L2p + 2Lf . Then

there exists a unique solution to (3.2).

Proof. We will use a contraction theorem on

Lβ = L2,β(Ω× [0, T ],F ⊗B([0, T ]), (A(ω, dt) + λ(dt))P(dω))×L2,β(p)×L2,β(W ).

We construct a mapping Γ that to each (P,Q,R) ∈ L2,β(A + λ) × L2,β(p) ×L2,β(W ) associates (Y, U, Z) solution to equation (3.7) when the generators are givenby ft(Pt, Qt) and gt(Pt, Rt). Such map is well defined: indeed if we fix (P,Q,R) ∈L2,β(A+λ)×L2,β(p)×L2,β(W ), thanks to assumption (3-B), the generators are knownprocess that satisfy assumption (3-B′) and proposition 3.6 and 3.9 give us the existenceand uniqueness of (Y, U, Z) ∈ L2,β(A+λ)×L2,β(p)×L2,β(W ). Notice that thanks tothe Lipschitz conditions on g and f , if we take two triplets (P ′, Q′, R′) ≡ (P ′′, Q′′, R′′)in L2,β(A + λ) × L2,β(p) × L2,β(W ), then fs(Y ′, U ′) ≡ fs(Y

′′, U ′′) in L2,β(A) andgs(Y

′.Z ′) ≡ gs(Y′′, Z ′′) in L2,β(W ).

Consider now (P ′, Q′, R′) and (P ′′, Q′′, R′′) in Lβ , and consider their images throughΓ, (Y ′, U ′, Z ′) = Γ(P ′, Q′, R′) and (Y ′′, U ′′, Z ′′) = Γ(P ′′, Q′′, R′′). Denote Y =Y ′ − Y ′′, P = P ′ − P ′′ and so on. Denote also ft = ft(P

′t , Q

′t) − ft(P

′′t , Q

′′t ) and

similarly denote g. (Y , U , Z, K) satisfies

Y =

∫ T

t

fsdAs +

∫ T

t

gsds−∫ T

t

∫E


t

ZsdWs + KT − Kt.

We now apply Ito’s Lemma to eβAseγsY 2s obtaining, after taking expectation,

βE[∫ T

0

eβAseγsY 2s dAs

]+ γE

[∫ T

0

eβAseγsY 2s ds

]+ E

[∫ T

0

eβAseγsZ2sdWs

]+ E

[∫ T

0

∫E

eβAseγsU2sφs(de)dAs

]≤ 2E

[∫ T

0

eβAseγsf 2s dAs

]+ 2E

[∫ T

0

eβAseγsg2sds

]+ 2E

[∫ T

0

Ys−dKs

].

As in the proof of proposition 3.9, we have that∫ T

0

Ys−dKs ≤ 0.

Denote by || · ||β,γ,A the norm (equivalent to || · ||L2,β(A))(E[∫ T

0

eβAseγsY 2s dAs

])1/2

,

and similarly denote the norms || · ||β,γ,p and || · ||β,γ,W . Using the Lipschitz propertiesof f and g this gives

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3.3. Reflected BSDE in the general case

β||Y ||2β,γ,A + γ||Y ||2β,γ,W + ||U ||2β,γ,p + ||Z||2β,γ,W ≤

≤ 2LfE[∫ T

0

eβAseγs|Ys||Ps|dAs]

+ 2LpE

[∫ T

0

eβAseγs|Ys|(∫

E

|Q2s|)1/2

dAs

]

+ 2LgE[∫ T

0

eβAseγs|Ys||Ps|ds]

+ 2LWE[∫ T

0

eβAseγs|Ys||Rs|ds].

Using the inequality 2ab ≤ αa2 + b2/α for a, b ≥ 0 we obtain:

β||Y ||2β,γ,A + γ||Y ||2β,γ,W + ||U ||2β,γ,p + ||Z||2β,γ,W

≤ Lf√αp||Y ||2β,γ,A + Lf

√α||P ||2β,γ,A +

L2p

α||Y ||2β,γ,A + α||Q||2β,γ,p

+Lg√α||Y ||2β,γ,W + Lg

√α||P ||2β,γ,W +

L2W

α||Y ||2β,γ,W + α||R||2β,γ,W .

Rewriting we obtain the following relation:

||U ||2β,γ,p + ||Z||2β,γ,W +

(β −

L2p

α− Lf√

α

)||Y ||2β,γ,A

+

(γ − L2

W

α− Lg√

α

)||Y ||2β,γ,W

≤ Lf√α||P ||2β,γ,A + α||Q||2β,γ,p + Lg

√α||P ||2β,γ,W + α||R||2β,γ,W . (3.28)

Since β > L2p + 2Lf , it is possible to choose α ∈ (0, 1) such that

β >L2p

α+

2Lf√α,

and for that α, choose γ such that γ > L2W/α + 2Lg/

√α. The relation (3.28) rewrites

as

Lf√α||Y ||2β,γ,A +

Lg√α||Y ||2β,γ,W + ||U ||2β,γ,p + ||Z||2β,γ,W

≤ Lf√α||P ||2β,γ,A + α||Q||2β,γ,p + Lg

√α||P ||2β,γ,W + α||R||2β,γ,W

= α

(Lf√α||P ||2β,γ,A + ||Q||2β,γ,p +

Lg√α||P ||2β,γ,W + ||R||2β,γ,W

). (3.29)

Now

Lf√α||P ||2β,γ,A +

Lg√α||P ||2β,γ,W = E

[∫ T

0

eβAseγsP 2s (Lf√αdAs +

Lg√αds)

]is a norm equivalent to ‖P‖L2,β(A+λ). We have thus that Γ is a contraction on L for theequivalent norm

‖(Y, U, Z)‖2Lβ ,γ =

Lf√α||Y ||2β,γ,A +

Lg√α||Y ||2β,γ,W + ||U ||2β,γ,p + ||Z||2β,γ,W .

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Since the space is complete, the contraction theorem assures us the existence of aunique triplet (Y, U, Z) in Lβ such that (Y, U, Z) = Γ(Y, U, Z), and (Y, U, Z,K) isthe solution to (3.2), where K is the one associated to (Y, Z, U) by the map Γ. Sincewe know

This last result generalizes the case of Brownian and Poisson noise, allowing for amore general structure in the jump part.

If we are interested only on a BSDE driven by a marked point process, the proofabove still applies when the filtration G is generated only by p and the data are adaptedto it. Then we have the counterpart of theorem 3.17

Theorem 3.18: Let assumptions (3-A) and (3-B)(i,ii,iv) hold for some β > L2p + 2Lf ,

but with the data adapted to the filtration G. Then the system (3.5) admits a uniquesolution in L2,β(A)× L2,β(p)× I2.

Proof. This is proven exactly as the case with also a Brownian motion. First, we showas in 3.6, the solution lies in L2,β(A)× L2,β(p)× I2 and, using Ito’s formula, that it isunique. Next we build a contraction on this space, and obtain existence and uniquenesswhen the generator depends on (Y, U).

Remark 3.19. A similar result does not hold in general in L1. Counter examples aregiven in Confortola, Fuhrman, and Jacod [20], where additional hypotheses are thenadded to obtain an existence and uniqueness result. We also refer to Confortola [18]where the case Lp is analysed.

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CHAPTER4Additional results on the reflected backward SDE

In this chapter we establish some results for the Reflected BSDE solved in the previouschapter. We first state and prove an approximation result when the generator depends onthe solution in a specific way. This result will be used both in the next chapter and hereto obtain a comparison theorem. It is well known that general comparison theoremsdo not hold in this framework with jumps, both for normal BSDE and reflected ones.Nevertheless, with additional hypotheses as in theorem 2.15, it is possible to comparethem. Some examples of papers in which a comparison result for reflected BSDE withjumps is established is Crepey and Matoussi [22], Essaky [36], Quenez and Sulem [74],and Ren and El Otmani [75], along many others.

The ingredients are again a terminal data ξ, two progressive processes fs and gs anda cadlag barrier process h, all satisfying the assumptions for existence and uniquenessof the RBSDE from chapter 3, as well as a P ⊗ E-measurable function ρ that satisfiesassumption (2-K) (and A must satisfy (2-K’)), we want to compare reflected BSDE ofthe following form through the use of non reflected BSDE as those defined in section2.3:

Yt = ξ +∫ TtfsdAs +

∫ TtUs(e)(ρs(e)− 1)φs(de)dAs +

∫ Ttgsds

−∫ TtUs(e)q(dsde)−


Yt ≥ ht∫ T0

(Yt− − ht−)dKt = 0.

(4.1)

The particular form of the equation might seem unusual now, but is the one that will beused in chapter 6 and more general than the one needed in chapter 5. The part added tothe f generator is Lipschitz in U with coefficient M ′, where M ′ = max|M − 1|, 1with M the bound for the function ρ. We thus assume the data ξ, h to satisfy assump-tion (3-B) and f, g to satisfy assumption (3-B′) for β > (M ′)2 Then this equation has

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Chapter 4. Additional results on the reflected backward SDE

one unique solution in(L2,β(A) ∩ L2,β(W )

)×L2,β(p)×L2β(W )×I2 for β > (M ′)2

thanks to theorem 3.17.

4.1 Approximating the RBSDE

To establish a comparison theorem, we first need to state an approximation result. Ba-sically we want to write the solution to the reflected BSDE as the limit of a sequence ofbasic BSDE for which we already have the comparison theorem 2.15. This is akin tothe the penalization method, but in our case we already know that the (limit) solutionexists.

We start by defining for all n ≥ 0 the following penalized BSDE

Y nt = ξ +

∫ T

t

fsdAs +

∫ T

t

∫E


∫ T

t

gsds

−∫ T

t

Uns (e)q(dsde)−

∫ T

t

Zns dWs +Kn

T −Knt , (4.2)

where Knt = n

∫ t0

(Y ns − hs)

− ds. We see that for each n this is a standard BSDE withgenerators fns (u) = fs +

∫E

(e)u(e)(ρs(e) − 1)φs(de) and gns (y) = gs + n(y − hs)−,thus existence and uniqueness in

(L2,β(A) ∩ L2,β(W )

)× L2,β(p) × L2β(W ) for β >

(M ′)2 is assured by theorem 2.14, where M ′ = max|M − 1|, 1. The comparisontheorem 2.15 tells us that Y n

t ≤ Y n+1t since all the hypotheses are verified (in particular

gn(y) ≤ gn+1(y) for all y ∈ R. We want to prove we can approximate (4.1) with (4.2).Let us start by some lemmas

Lemma 4.1: Let Y n be the solution to 4.2. Then there exists a Y such that Y nt Yt ≤

Yt, where Y is the solution to the RBSDE 4.1.

Proof. Let us start by noticing that the kernel ρ appearing in (4.1) introduces, as de-scribed in section 2.1.2, a probability Pρ P. This also holds if instead of ρ we useρ + ε, as long as ε > 0 is small enough (let us say smaller than some ε > 0) suchthat assumptions (2-K) and (2-K’) are satisfied. The difference is that in this case thekernel ρ + ε is bounded from below, and the introduced probability Pρ+ε is equivalentto the reference probability P. Thus we fix ε such that η > 3 + (M ′ + ε)4, where η isthe parameter appearing in assumption (2-K’). Then it is possible to show that for any0 < ε < ε Y satisfies

Yt + εΓU = ess supτ≥t

Eρ+ε

[∫ τ∧T

t

fsdAs +

∫ τ∧T

t


∣∣∣∣ Ft] ,(4.3)

where

ΓU = ess supτ≥t

Eρ+ε

[∫ τ

t

∫E

Us(e)φs(de)dAs

∣∣∣∣ Ft] .As already said, we have that Y n ≤ Y n+1 thanks to the comparison theorem. Now

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4.1. Approximating the RBSDE

for fixed n consider Y n between t and a generic stopping time τ , we have

Y nt +ε

∫ τ

t

∫E

Uns (e)φs(de)dAs = Eρ+ε

[Y nτ +

∫ τ

t

fsdAs +

∫ τ

t

gsds+Knτ −Kn

t

∣∣∣∣ Ft]≥ Eρ+ε

[Y nτ ∧ hτ1τ<T + ξ1τ≥T +

∫ τ

t

fsdAs +

∫ τ

t

gsds

∣∣∣∣ Ft] . (4.4)

Define now τ ∗t = infs ≥ t : Kns −Kn

t > 0 ∧ T . Take ω such that τ ∗t (ω) < T . Then∃tk τ ∗t (ω) such that Y n

tk(ω) ≤ htk(ω). Then Y n

τ∗t(ω) ≤ hτ∗t (ω), since Y n and h are

cadlag. ThenY nτ∗t1τ∗t <T = Y n

τ∗t∧ hτ∗t 1τ∗t <T

and

Y nt + ε

∫ τ∗t

t

∫E

Uns (e)φs(de)dAs = Y n

τ∗t∧ hτ∗t 1τ∗t <T + ξ1τ∗t ≥T +

∫ τ∗t

t

gsds

+

∫ τ∗t

t

fsdAs +

∫ τ∗t

t

∫E

Uns (e)(ρs(e) + ε− 1)φs(de)dAs

−∫ τ∗t

t

∫E

Uns (e)q(dsde)−

∫ τ∗t

t

Zns dWs. (4.5)

By taking (ρ+ ε)−expectation conditional on Ft we obtain, together with (4.4)

Y nt +εΓU

n

= ess supτ≥T

Eρ+ε

[Y nτ ∧ hτ1τ<T + ξ1τ≥T +

∫ τ∧T

t

fsdAs +

∫ τ∧T

t

gsds

∣∣∣∣ Ft] .Where

ΓUn

= ess supτ≥t

Eρ+ε

[∫ τ

t

∫E

Us(e)φs(de)dAs

∣∣∣∣ Ft]Comparing this last to (4.3) we notice that Pρ+ε-a.s Y n

t +εΓUn ≤ Yt+εΓ

U . Since ΓUn is

non-negative, this also means Y nt ≤ Yt+εΓ

U . Since the sequence Y n is non-decreasingwe have the existence of a limit Y such that

Y nt Yt ≤ Yt + εΓU .

The last inequality holds Pρ+ε and P almost surely as they are equivalent. Now we justhave to show that ΓU is bounded so we can send ε to zero and prove the lemma. To thisend notice that

|ΓU | ≤∣∣∣∣ess sup

τ≥tEρ+ε

[∫ τ

t

∫E

Us(e)φs(de)dAs

∣∣∣∣ Ft]∣∣∣∣≤ Eρ+ε

[∫ T

0

∫E

|Us(e)|φs(de)dAs∣∣∣∣ Ft]

=E[LεT∫ T

0


∣∣∣ Ft]Lεt

≤ E [(LεT )2| Ft](Lεt)

2+

E[(∫ T

0


)2∣∣∣∣ Ft]

4

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Where Lε denotes the Girsanov martingale. The last term in the right hand side is P-a.s.finite as we have by Cauchy-Schwarz inequality

(∫ T

0

∫E

e−βAs/2eβAs/2|Us(e)|φs(de)dAs)2

≤∫ T

0

e−βAsdAs

∫ T

0

∫E


≤ 1

β

∫ T

0

∫E

eβAs|Us(e)|2φs(de)dAs,

where the last term is in L1(P). As for the first term on the right hand side we have thatit is bounded by E

[eηAT

∣∣ Ft] thanks to (2.8) , which is also P-a.s. finite too. Then ΓU

is bounded by a random variable that does not depend on ε and we conclude by sendingε to zero.

The next step is establishing a lemma on the norms of the solutions to the penalizedequations:

Lemma 4.2: Let (Y n, Un, Zn) be the solution to the penalized BSDE (4.2). Then thereexist a constant Cp depending on ξ, f, g, h but independent of n such that

E[∫ T

0

eβAs|Y ns |2(dAs + ds)

]+ E

[∫ T

0

∫E

eβAs|Uns (e)|2φs(de)dAs

]+ E

[∫ T

0

eβAsZ2sds

]< Cp.

Proof. The bound is obtained by applying Ito’s formula to eβ(At+t)(Y nt )2 and pro-

ceeding like in the proof of 3.6, with the difference that there is the additional term∫ T0

∫EUns (e)φs(de)dAs. In the computation appears the term

∫ T

0

eβ(As+s)Y ns dK

ns =

∫ T

0

eβ(As+s)Y ns n(Y n

s − hs)−ds ≤∫ T

0

eβ(As+s)hsdKns ,

where the last inequality is due to the fact that the integrand is non zero only when

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4.1. Approximating the RBSDE

hs ≥ Y ns . By applying Ito’s formula between 0 and T and taking expectation we obtain

βE[∫ T

0

eβ(As+s)(Y ns )2dAs

]+ E

[∫ T

0

∫E

eβ(As+s)(Uns (e))2φs(de)dAs

]+ βE

[∫ T

0

eβ(As+s)(Y ns )2ds

]+ E

[∫ T

0

eβ(As+s)(Zns )2ds

]≤ E

[eβ(AT+T )|ξ|2

]+ 2E

[∫ T

0

eβ(As+s)Y ns fsdAs

]+ 2E

[∫ T

0

eβ(As+s)Y ns gsds

]+ 2E

[∫ T

0

eβ(As+s)hsdKns

]+ 2E

[∫ T

0

eβ(As+s)Y ns U

ns (e)(ρs(e)− 1)φs(de)dAs

]≤ E

[eβ(AT+T )ξ2

]+

1

αE[∫ T

0


]+

1

αE[∫ T

0

eβ(As+s)(Y ns )2ds

]+ αE

[∫ T

0

eβ(As+s)f 2s dAs

]+

1

1 + aE[∫ T

0

∫E

eβ(As+s)(Uns (e))2φs(de)dAs

]+ αE

[∫ T

0

eβ(As+s)g2sds

]+ (M ′)2(1 + a)E

[∫ T

0


]+ γE

[supte(β+δ)(At+t)h2

t

]+

1

γE

[(∫ T

0

e(β−δ)As+s2 dKn

s

)2],

(4.6)

The last term is estimated again by considering the dynamic of e(β−δ) (At+t)2 Y n

t obtainingthe following bound

E

[(∫ T

0

e(β−δ)As+s2 dKn

s

)2]≤ 9E

[eβ(AT+T )|ξ|2

]+ 9E

[(Y n

0 )2]

+9(β − δ)2

4δE[∫ T

0

eβ(As+s)Y 2s ds

]+

9(β − δ)2

4δE[∫ T

0

eβ(As+s)Y 2s dAs

]+

9

δE[∫ T

0

eβ(As+s)f 2s dAs

]+

9

δE[∫ T

0

eβ(As+s)g2ss

]+ 9E

[∫ T

0

eβ(As+s)Z2sds

]+ 9E

[∫ T

0

∫E


]+

9(M ′)2

δE[∫ T

0

∫E


].

Notice that E [(Y n0 )2] ≤ E [max(Y 0

0 )2, (Y0)2]. By plugging this last bound in (4.6),and choosing a small enough and α, γ big enough, we have proven the theorem.

Proposition 4.3: We have that Yt = Yt and thus Y nt Yt.

Proof. First notice that by Lebesgue dominated convergence theorems we have that

Y n L2,β(A)→ Y . Thanks to lemma 4.2, we know that (Un, Zn) are bounded in L2,β(p) ×

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L2,β(W ) for β > (M ′)2. Then thanks to proposition 2.16 we know that there exist(U , Z, K) that solve

Yt = ξ +

∫ T

t

fsdAs +

∫ T

t

gsds+

∫ T

t

∫E

Us(e)(ρs(e)− 1)φs(de)dAs

−∫ T

t

∫E


t

ZsdWs + KT − Kt. (4.7)

Now it holds that Yt ≥ ht. Indeed if we take (4.2) between 0 and T and we takeexpectation:

nE[∫ T

0

(Y ns − hs)−ds

]=

E[Y n0 ]− E

[ξ −

∫ T

0

(fs

∫E

U − sn(e)(ρs(e)− 1)φs(de)

)dAs −

∫ t

0

gsds

],

and, dividing by n, E[∫ T

0(Y n

s − hs)−ds]→ 0, where we used the fact that

E∫ T

0

∫E

Uns (e)(ρs(e)− 1)φs(de)dAs ≤M

from lemma 4.2. Then

E[∫ T

0

(Ys − hs)−ds]

= E[∫ T

0

limn

(Y ns − hs)−ds

]≤ lim inf

nE[∫ T

0

(Y ns − hs)−ds

]= 0.

Then P-a.s.∫ T

0(Y n

s − hs)−ds = 0 and since the process are cadlag, P-a.s. Ys ≥ hs forall t < T . Since YT = ξ, this holds also for T . This means that

Yt +

∫ t

0

fsdAs +

∫ t

0

gsds ≥ ξ1t≥T + ht1t<T +

∫ t

0

fsdAs +

∫ t

0

gsds. (4.8)

We introduce again (see the proof of lemma 4.1) the equivalent probability Pρ+ε ∼ Pthrough a Girsanov transform with kernel ρ+ ε. Define the non-negative quantity

ΓUt = ess supτ≥t

Eρ+ε

[∫ τ

t

∫E

Us(e)φs(de)dAs

∣∣∣∣ Ft] .If we consider equation (4.7) between t and a stopping time τ and add

ε

∫ τ

t

∫E

Us(e)φs(de)dAs

to both sides, we obtain after taking essup over all stopping times on both sides that

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4.2. A comparison theorem

Yt +

∫ t

0

fsdAs +

∫ t

0

gsds+ εΓUt = ess supτ≥t

ψτ − Kt,

where

ψt = ξ +

∫ t

0

fsdAs +

∫ t

0

gsds+ Kt.

Thus

Yt +

∫ t

0

fsdAs +

∫ t

0

gsds+ εΓUt (4.9)

is a Pρ+ε-supermartingale as it is the difference of a supermartingale (it is a Snell enve-lope) and an increasing process. Notice that since Pρ+ε ∼ P, the inequality (4.8) alsoholds Pρ+ε-a.s. Also if we add ΓUt to the left hand side of (4.8), the inequality is stilltrue. Then (4.9) is a Pρ+ε-supermartingale that dominates

ψt = ξ1t≥T + ht1t<T +

∫ t

0

fsdAs +

∫ t

0

gsds.

On the other hand we have that (see the proof of lemma 4.1)

Yt + εΓU = Yt + ε ess supτ≥t

Eρ+ε

[∫ τ

t

∫E

Us(e)φs(de)dAs

∣∣∣∣ Ft]is the Pρ+ε Snell envelope of the same quantity ψt. As the Snell envelope is the smallestsupermartingale that dominates ηt, it holds that Pρ+ε-a.s.

Yt + εΓU ≤ Y + εΓU .

But since the two probabilities are equivalent, this means that the inequality holds alsoP-a.s. Now we have already shown in the proof of lemma 4.1 that

ΓUt ≤ E[eηAt

∣∣ Ft]+E[∫ T

0

∫EeβAsU2

s (e)φsdedAs

∣∣∣ Ft]4β

.

The same holds for ΓUt . Thus by sending ε to zero we obtain that Yt ≤ Yt P-a.s. Sincethe processes are cadlag this holds up to indistinguishability. Confronting it with lemma4.1 we obtain that Yt = Yt.

4.2 A comparison theorem

Thanks to the approximation result, we can provide a comparison theorem for reflectedBSDE of the form (4.1).

Theorem 4.4: Assume (2-K) holds for ρ and (2-K’) holds for A. We are given two setsof data ξi, f i, gi, hi satisfying assumptions (3-B)-(i)(iv) and (3-B′) for some β > (M ′)2,and such that ξ1 ≤ ξ2, f 1

s ≤ f 2s , g1

s ≤ g2s and h1

s ≤ h2s. Then we have that the respective

solutions (Y i, U i, Zi, Ki) are such Y 1t ≤ Y 2

t for all t ∈ [0, T ], P-almost surely.

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Proof. We can approximate each RBSDE with the BSDE defined above. That is fori = 1, 2:

Y i,nt = ξi +

∫ T

t

f isdAs +

∫ T

t

gisds+

∫ T

t

∫E

U i,ns (ρs(e)− 1)φs(de)dAs

−∫ T

t

U i,ns (e)q(dsde)−

∫ T

t

Zi,ns +Ki,n

T −Ki,nt

with Ki,nt = n

∫ t0(Y i,n

s − his)−ds. Remember that Y i,nt Y i

t thanks to proposition 4.3.Since h1

s ≤ h2s, we have that (y − h1

s)− ≤ (y − h2

s)− and we can apply the comparison

theorem 2.15 and obtain that

Y 1,nt ≤ Y 2,n

t ∀t ∈ [0, T ], P-a.s. ∀n ≥ 1.

By taking the limit for n→ +∞, we obtain that Y 1t ≤ Y 2

t ∀t ∈ [0, T ], P-a.s. .

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CHAPTER5Randomization technique for Optimal stopping

and constrained BSDE

In this chapter we want to extend the results of Fuhrman, Pham, and Zeni [41] to thecase of a BSDE where the driving noises are both a Wiener process and a randommeasure associated to a Marked Point Process. The idea is to replace the stoppingtime in an optimal stopping problem with an artificial randomizing single jump pointprocess. A dual control problem is introduced, where the expected reward is maximizedover a set of possible changes of intensity of said process. The value function of thisartificial problem is represented through the use of a constrained BSDE, linked to thereflected BSDE from chapter 3. The extension to our case is pretty straightforward,but it is important to take care and verify some technical details. In particular one hasto check that the change of intensity does not touch the intensity of the marked pointprocess already present. The main object studied is the constrained BSDE

Yt = ξ1η≥T +

∫ T

t

fs1[0,η]dAs +

∫ T

t

gs1[0,η]ds−∫ T

t

∫E

Us(e)q(dsde)

−∫ T

t

ZsdWs +

∫ T

t

hsdRs −∫ T

t

VsdRs + KT − Kt ∀t ∈ [0, T ],

(5.1)

With the constraintVt ≤ 0.

This kind of randomization technique was first introduced in Kharroubi et al. [59],where the authors randomize the control in a impulse control problem in the diffusivecase by replacing the control strategy with a point process. This leads them to write aBSDE equation driven both by a Wiener process and a jump process, with a constraint

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Chapter 5. Randomization technique for Optimal stopping and constrained BSDE

not in the Y part but rather in the jump part U appearing in the martingale. This equa-tion is used to represent the value function of the impulse control problem. The sameapproach has then been used in a variety of settings, often related to optimal control.A non exhaustive list of examples is Elie and Kharroubi [32], Elie and Kharroubi [33],Bandini and Fuhrman [3],Choukroun, Cosso, and Pham [14]. In particular in Khar-roubi and Pham [58] the authors obtain a representation of viscosity solutions to fullynon-linear Hamilton Jacobi Bellman equations through the use of a constrained BSDE.One of the main advantage of using this class of equations is that they can be simulated,as seen in Kharroubi, Langrene, and Pham [56, 57].

In most of the cases above the existence of a solution is obtained through a penal-ization argument, but in our case, as in Fuhrman, Pham, and Zeni [41], a solution isconstructed directly from the solution to the reflected BSDE. In order to link the con-trained BSDE to the artificial control problem, we still need to use an approximation ofthe reflected BSDE obtained through penalization as in chapter 4.

5.1 The constrained BSDE

We start again with the framework described in section 2.2. Denote by Tt the set of allstopping times greater than t.

Under the assumptions (3-A) and (3-B), the following reflected equation admits aunique solution (Y, U, Z,K) as seen in chapter 3:

Yt = ξ +



∫ Tt

∫EUs(y)q(dsdy)


Yt ≥ ht, ∀t ∈ [0, T ] a.s.∫ T0


1Yt−=ht−∀t ∈ [0, T ] a.s..

(5.2)

Consider now the case when f and g do not depend on Y, U, Z but are known processes,and assumption (3-B′) is in force instead of (3-B)(ii)(iii). In this case the solutionrepresents the value function of an optimal stopping problem, precisely

Yt = ess supτ∈Tt

E[∫ τ

t

fsdAs +

∫ τ

0

gsds+ hτ1τ<T + ξ1τ=T

∣∣∣∣ Ft] .Consider now a second complete probability space (Ω′,F ′,P′), on which is defined

an exponential random variable η ∼ exp(1). This can be viewed as a one jump countingprocess where the counting process is Rt = 1η≤t and the compensator is Dt = t ∧ η.

We can then construct the space (Ω, F , P) by setting Ω = Ω×Ω′, F = F ⊗F ′ andP = P ⊗ P′. All the elements p,W, η have a natural extension to the space (Ω, F , P).The spaces L2,β(A), L2,β(p), L2,β(W ) and I2, defined in section 2.2 have a naturalextension and we denote them in the same way. We also define the space L2,β(D) asthe following space of equivalence classes

L2,β(D) =

V : Ω× [0, T ]→ R, F-predictable, ‖V ‖2

L2,β = E∫ T

0

eβAs |Vs|2dDs <∞.

We note that ‖V ‖2L2,β(D)

= E∫ T

0eβAs|Vs|2dRs.

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5.1. The constrained BSDE

On (Ω, F , P) p, W and η are independent, W is still a Brownian motion and thecompensator of p(dsde) is still φs(de)dAs:

Lemma 5.1: The compensator of p(dsde) under P is φs(de)dAs. Moreover P(Tn =η) = 0 for all n.

Proof. We need to show that for a non-negative P(F) ⊗ E-measurable process C itholds that

E[∫ T

0

∫E

Ct(e)p(dtde)

]= E

[∫ T

0

∫E

Ct(e)φt(de)dAt

]To this end first consider s, t such that s < t, and consider two sets G ∈ Fs and

G′ ∈ F ′s, as well as a set H ∈ E . We have

E[∫ T

0

∫E

1(s,t](r)1G(ω)1G′(ω′)1H(e)p(ω, drde)

]= E

[∫ T

0

∫E

1(s,t](r)1G(ω)1H(e)E′ [1G′(ω′)] p(ω, drde)]

The term E′ [1G′(ω′)] is deterministic, and the process 1(s,t](r)1G(ω)1H(e) is non-negative and P(F) ⊗ E-measurable, and thus by the definition of F-compensator ofp this means:

E[∫ T

0

∫E

1(s,t](r)1G(ω)1H(e)E′ [1G′(ω′)] p(ω, drde)]

= E[∫ T

0

∫E

1(s,t](r)1G(ω)1H(e)E′ [1G′(ω′)]φr(ω, de)dAr(ω)

]and again

E[∫ T

0

∫E

1(s,t](r)1G(ω)1H(e)E′ [1G′(ω′)]φr(ω, de)dAr(ω)

]= E

[∫ T

0

∫E

1(s,t](r)1G(ω)1G′(ω′)1H(e)φr(ω, de)dAr(ω)

].

Then by a monotone class argument this means that, for fixed s, t and H ∈ E , for allF ∈ Fs it holds

E[∫ T

0

∫E

1(s,t](r)1f (ω)1H(e)φr(ω, de)dAr(ω)

]= E

[∫ T

0

∫E

1(s,t](r)1F (ω)1H(e)φr(ω, de)dAr(ω)

].

Using again the monotone class theorem, we have this relation holds for all non-negative P(F)⊗ E-measurable process C and the lemma is proven. As for the secondpoint, we have that

P(η = Tn) = E[P(η = t)|t=Tn

]= 0.

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Of course, the compensator of R is still D.We define now what is commonly called a randomized BSDE with a constraint.

Yt = ξ1η≥T +

∫ T

t

fs1[0,η]dAs +

∫ T

t

gs1[0,η]ds−∫ T

t

∫E

Us(e)q(dsde)

−∫ T

t

ZsdWs +

∫ T

t

hsdRs −∫ T

t

VsdRs + KT − Kt ∀t ∈ [0, T ],

(5.3)

With the constraintVt ≤ 0 dDt(ω)P(ω)− a.s. (5.4)

A solution is a quintuple (Y , U , Z, V , K) in (L2,β(A)∩L2,β(W ))×L2,β(p)×L2,β(W )×L2,β(R)× I2 such that (5.3) and (5.4) are satisfied. We look for a minimal solution, inthe sense that if another quintuple (Y ′, U ′, Z ′, V ′, K ′) satisfies it, then Yt ≤ Y ′t for allt ∈ [0, T ]. A solution can be constructed starting from a solution to (3.2).

Proposition 5.2: There exists a solution to (5.3)-(5.4). It can be defined by setting, forω = (ω, ω′), t ∈ [0, T ]:

Yt(ω) = Yt(ω)1t<η(ω′), Ut(ω, e) = Ut(ω, e)1t≤η(ω′), Zt(ω) = Zt(ω)1t≤η(ω′)(5.5)

Vt(ω) = (ht(ω)− Yt(ω))1t≤η(ω′), Kt(ω) = Kt∧η(ω′)(ω).

Remark 5.3. Remember that integrals of the form∫ ba

denote by convention∫

(a,b].

Proof. It is clear from the definition (5.5) that (Y , U , Z, V , K) is in (L2,β(A)∩L2,β(W ))× L2,β(p)× L2,β(W )× L2,β(D)× I2 with underlying space (Ω, F , P). Thanks to thereflecting condition, the constraint Vt ≤ 0 holds. It is enough to check it on the threedisjoint sets η > T, 0 ≤ t < η < T and 0 < η < T, η ≤ t ≤ T. Notice thatthese three sets form, P-a.s. a partition of Ω. On η > T we have Yt = Yt, Ut = Ut,Zt = Zt, Kt = Kt. This entails that

∫ Tt

∫EUs(e)q(dsde) =

∫ Tt

∫EUs(e)q(dsde) and∫ T

tZsdWs =

∫ TtZsdWs, thanks to the local property of the stochastic integral. On

0 ≤ t < η < T equation (5.3) becomes

Yt =

∫ η

t

fsdAs +

∫ η

t

gsds−∫ T

t

∫E

Us(e)q(dsde)

−∫ T

t

ZsdWs + hη − Vη + KT − Kt.

Since we are on 0 ≤ t < η < T, it holds that Yt = Yt,∫ Tt

∫EUs(e)q(dsde) =∫ η

t

∫EUs(e)q(dsde),

∫ TtZsdWS =

∫ ηtZsdWs, KT = Kη and Kt = Kt. Thus, the

previous equation becomes

Yt =

∫ η

t

fsdAs +

∫ η

t

gsds−∫ η

t

∫E

Us(e)q(dsde)−∫ η

t

ZsdWs + hη − Vη +Kη −Kt,

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5.2. The dual problem

that, remembering that hη − Vη = Yη, holds thanks to (3.2). On the last set 0 < η <T, η ≤ t ≤ T, we have Yt = 0 and all the integrals are null: indeed∫ T

t

∫E

Us(e)q(dsde) =

∫(t,T ]

∫E

Us(e)1s≤ηq(dsde) = 0

since η ≤ t, and the same applies to the rest of integrals. Equation (5.3) reduces then to

0 = 0 +Kη −Kη = 0,

which is of course satisfied. The minimality property will be proved later in proposition5.7.

5.2 The dual problem

We introduce now what is usually called a “Dual Control Problem”, and with it a gainfunctional to maximise and a relative value function. The idea behind is that the controlis randomized: instead of choosing a stopping time, we introduce a random variable ηas presented before. At this point we maximize over the possible changes of intensityof the one jump process associated to η. We introduce the following set of functions:

V =ν : Ω× [0,+∞)→ (0,+∞), F-predictable and bounded

For each ν ∈ V we consider the following supermartingale

Lνt = e∫ t0 log(ηs)dRs+

∫ t0 (1−νs)dDs = (1t<η + νη1t≥η)e

∫ t∧η0 (1−νs)ds

For all ν ∈ V , Lν is a F-martingale on [0, T ] since ν is bounded. Then we can definea new probability Pν such that

dPν

dP

∣∣∣∣Ft

= Lνt

Under Pν , the compensator of R and p change in the following way:

Lemma 5.4: Fix ν ∈ V . Under Pν , the compensator of Rt is∫ t

0νsdDs and the com-

pensator of p(dsde) is still φs(de)dAs.

Proof. We have to show, that for any positive P ⊗ E-measurable process C and for anypositive P-measurable process H it holds that

Eν[∫ T

0

∫E

Cs(e)p(dsde)

]= Eν

[∫ T

0

∫E

Cs(e)φs(de)dAs

]Eν[∫ T

0

HsdRs

]= Eν

[∫ T

0

HsνsdDs

].

Let us show it for p(dsde). The left hand side is, using Dellacherie [23, T47,Chapter

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4],

Eν[∫ T

0

∫E

Cs(e)p(dsde)

]= E

[LνT

∫ T

0

∫E

Cs(e)p(dsde)

]= E

[∫ T

0

∫E

LνsCs(e)p(dsde)

]= E

[ ∑0<Tn≤T

CTn(ξn)LνTn

]

= E

[ ∑0<Tn≤T

CTn(ξn)LνT−n

]

= E[∫ T

0

∫E

Cs(e)Lνs−p(dsde)

], (5.6)

since LνTn = LνT−n

P-a.s. thanks to lemma 5.1. The right hand side, with a similarreasoning gives:

Eν[∫ T

0

∫E

Cs(e)φs(de)dAs

]= E

[LνT

∫ T

0

∫E

Cs(e)φs(de)dAs

](5.7)

= E[∫ T

0

∫E

Cs(e)Lνs−φs(de)dAs

]. (5.8)

Now, Cs(e)Lνs− is a positive P ⊗ E-measurable process and, under P, we know thanksto lemma 5.1 that φs(de)dAs is the compensator of p(dsde). Thus the last terms areequal and we proved the claim. As for the compensator of R, it is just sufficient to notethat Lνη = νηL

νη− and repeat a similar reasoning:

Eν[∫ T

0

HsdRs

]= E

[LνT

∫ T

0

HsdRs

]= E

[∫ T

0

HsLνsdRs

]= E

[HηL

νη

]= E

[HηνηL

νη−

]= E

[∫ T

0

HsLs−νsdRs

]= E

[∫ T

0

HsLs−νsdDs

]= E

[LT

∫ T

0

HsνsdDs

]= Eν

[∫ T

0

HsνsdDs

],

where in the third line we used the fact thatHsLs−νs is a positive P-measurable processand under P the compensator of R is D. This concludes the proof.

We can now finally define the dual control problem. For each ν in V we considerthe following quantity

J(t, ν) = Eν[ξ1η≥T +

∫ T∧η

t∧ηfsdAs +

∫ T∧η


∣∣∣∣ Ft] ,and the value function of our problem defined as

v(t) = ess supν∈V

Eν[ξ1η≥T +

∫ T∧η

t∧ηfsdAs +

∫ T∧η


∣∣∣∣ Ft] . (5.9)

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We can then obtain the important result of this section, that the solution to the con-strained BSDE represents the value function of the problem

Theorem 5.5: Let Y be the solution to the constrained BSDE (5.3)-(5.4). Then for allt ∈ [0, T ], Yt = vt P-a.s.

We immediately have the following corollary thanks to the fact that Y0 = Y0 a.s.

Corollary 5.6: The value function to the natural optimal stopping problem and thevalue function of the dual problem coincide at t = 0, that is v(0) = v(0).

Poof of theorem 5.5. Let (Y , U , Z, V , K) be a solution to (5.3)-(5.4). We have that

Yt ≥ ξ1η≥T +

∫ T

t

fs1[0,η]dAs +

∫ T

t

1[0,η]ds+

∫(t,T ]

hsdRs

−∫ T

t

∫E


t

ZsdWs,

thanks to the fact that V is non-positive and K is increasing. Notice that∫

(t,T ]hsdRs =

hη1t<η≤T = hη1t<η<T since η 6= T a.s. Now for a fixed ν ∈ V , we can take ex-pected value with respect to Pν introduced above and obtain, thanks to the fact that un-der Pν W is still a Brownian motion and the compensator of p(dsde) is still φs(de)dAs,the following inequality:

Yt ≥ Eν[ξ1η≥T +

∫ T∧η

t∧ηfsdAs +

∫ T∧η


∣∣∣∣ Ft] .Taking the ess sup on the right hand side, we obtain the first inequality

Yt ≥ v(t). (5.10)

Notice that we did not need it to be the minimal solution or the one defined in proposi-tion 5.2.

To prove the reverse inequality, we need to introduce a sequence of BSDE con-verging to our constrained BSDE defined in proposition 5.2. First, recall that thanks toproposition 4.3, the solution to the reflected BSDE can be approximated by the solutionof:

Y nt = ξ +

∫ T

t

fsdAs +

∫ T

t

gsds−∫ T

t

∫E

Uns (e)q(dsde)

−∫ T

t

Zns dWs + n

∫ T

t

(Y ns − hs)−ds. (5.11)

Starting from this last equation, it is possible to define an approximating sequence ofBSDE for the BSDE with constraint (5.3)-(5.4). Consider the following BSDE

Y nt = ξ1η≥T +

∫ T

t

fs1[0,η]dAs +

∫ T

t

gs1[0,η]ds−∫ T

t

∫E

Uns (e)q(dsde)

−∫ T

t

Zns dWs −

∫(t,T ]

V ns dRs +

∫ T

t

hsdRs + n

∫ T

t

(V ns )+

1[0,η]ds, t ∈ [0, T ]. (5.12)

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A unique solution (Y n, Un, Zn, V n) ∈ (L2,β(A) ∩ L2,β(W )) × L2,β(p) × L2,β(W ) ×L2,β(D) to this equation can be constructed by setting

Y nt (ω) = Y n

t (ω)1t<η(ω′), Unt (ω, e) = Un

t (ω, e)1t≤η(ω′)

Znt (ω) = Zn

t (ω)1t≤η(ω′), V nt (ω) = (ht(ω)− Y n

t (ω))1t≤η(ω′).

It is clear that this solution is in (L2,β(A)∩L2,β(W ))×L2,β(p)×L2,β(W )×L2,β(D).Using the same reasoning as in the proof of proposition 5.2, we can show that it solves(5.12).

Define now the following subset of V:

Vn = ν ∈ V : ν ≤ n , (5.13)

as well as another dual control problem

vn(t) = ess supν∈Vn

Eν[ξ1η≥T +

∫ T∧η

t∧ηfsdAs +

∫ T∧η


∣∣∣∣ Ft] .(5.14)

We claim that for all t ∈ [0, T ],

Y nt = vn(t) P-a.s.

Indeed consider a generic ν ∈ Vn and its associated probability Pν . Again, W is stilla Brownian motion and the compensator of p(dsde) is still φs(de)dAs, we obtain thefollowing relation by conditioning with respect to Ft under Pν

Y nt = Eν

[ξ1η≥T +

∫ T∧η

t∧ηfsdAs +

∫ T∧η

t∧ηgsds+

∫(t,T ]

hsdRs

∣∣∣∣ Ft]− Eν

[∫(t,T ]

V ns dRs

∣∣∣∣ Ft]+ Eν[n

∫ T

t

(V ns )+

1[0,η](s)ds

∣∣∣∣ Ft] .Under Pν , the compensator of Rt is

∫ t0νsdDs =

∫ T0νs1[0,η]ds we have that

Eν[∫

(t,T ]

V ns dRs

∣∣∣∣ Ft] = Eν[∫

(t,T ]

V ns 1[0,η]ds

∣∣∣∣ Ft]and the relation above becomes

Y nt = Eν

[ξ1η≥T +

∫ T∧η

t∧ηfsdAs +

∫ T∧η

t∧ηgsds+

∫(t,T ]

hsdRs

∣∣∣∣ Ft]+ Eν

[∫ T

t

(n(V ns )+ − νsV n

s )1[0,η](s)ds

∣∣∣∣ Ft] . (5.15)

Since for every ν ∈ Vn and for any real number x ∈ R it holds that nx+ − νx ≥ 0, itholds that for every ν ∈ Vn

Y nt ≥ Eν

[ξ1η≥T +

∫ T∧η

t∧ηfsdAs +

∫ T∧η

t∧ηgsds+

∫(t,T ]

hsdRs

∣∣∣∣ Ft] , (5.16)

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and thus Y nt ≥ vn(t). On the other hand, we can define for every 0 < ε ≤ 1 the quantity

νεt = n1Unt >0+ ε1−1≤Unt ≤0 − ε(Unt )−1

1Unt <−1. It clearly holds that νε ∈ Vn and

that n(Unt )+ − Un

t νεt ≤ ε. By setting ν = νε in the relation (5.15), we obtain

Y nt ≤ Eνε

[ξ1η≥T +

∫ T∧η

t∧ηfsdAs +

∫ T∧η

t∧ηgsds+

∫(t,T ]

hsdRs

∣∣∣∣ Ft]+ εT

≤ vn(t) + εT

and, by taking the limit as ε → 0, we obtain Y nt ≤ vn(t) which together with (5.16)

tells us that Y nt = vn(t) P-a.s.

Since Y nt Yt, we also have that Y n

t Yt for all t P-a.s. Since Vn ⊂ V , we alsohave that vn(t) ≤ v(t) and thus Y n

t ≤ vt. By taking the limit we obtain that for all t,Yt ≤ v(t) P-a.s. The opposite inequality also holds as seen above, and this concludesthe proof.

We can now prove the minimality of the solution defined in 5.2.

Proposition 5.7: The solution (Y , U , Z, V , K) to the constrained BSDE defined in 5.2is minimal.

Proof. Let (Y ′, U ′, Z ′, V ′, K ′) be another solution to (5.3)-(5.4). We know from thefirst part of the proof of theorem 5.5 that for all t ∈ [0, T ], Y ′ ≥ v(t) P-a.s.. Alsofrom theorem 5.5 we know that for all t ∈ [0, T ], Yt = v(t) P-a.s.. Thus for allt ∈ [0, T ] Yt ≤ Y ′t P-a.s. Since both Y ′ and Y are cadlag, this relation holds up toindistinguishability.

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CHAPTER6A switching problem for marked point processes

In this chapter we formulate a switching problem for marked point processes (and adiffusive part), and solve it using a system of reflected backward SDE with intercon-nected obstacles. The first peculiarity is how the switching problem is formulated. Thecurrent switching mode affects the running and final rewards, but also the dynamic ofthe point process by changing its compensator and thus its law. The formulation iswhat could be called “weak formulation”: every given switching strategy introducesa different probability, induced by the change in the compensator of the marked pointprocess. We seek thus to maximise the expected reward minus the switching costs overall possible strategies, but the law under which we take expectation depends on thestrategy. To tackle this problem, we make use of a system of reflected BSDE that carrythe information for the change of compensator.

Switching problems have attracted a lot of attention in the years. As we mentionedin section 2.4, an early formulation of the problem is given in Brennan and Schwartz[11]. There is a large number of works in which the switching problem or variationsof it have been treated, for example Brekke and Øksendal [9], Carmona and Ludkovski[12], Ludkovski [64], and Tang and Yong [79] among others.

The use of system of reflected BSDE to tackle this type of problem is fairly recent.The first paper to use this technique is Hamadene and Jeanblanc [49], where the authorssolve the problem with two possible switching modes and the diffusive dynamic of theunderlying process not dependent on the strategy. In order to do this they use the Snellenvelope characterization of the solution processes, as well as a doubly reflected BSDEto prove the existence of said solution. This approach is later generalized to the caseof any finite number of modes, first in Djehiche, Hamadene, and Popier [26] usingthe Snell envelope characterization, and then both in Hamadene and Zhang [50] andHu and Tang [53], where they establish existence of a system of reflected BSDE with

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Chapter 6. A switching problem for marked point processes

interconnected obstacles. In the latter, they solve the switching problem in the casewhere the drift of the underlying process depends on the mode. The generalization tothe case where the mode also affects the volatility, using a system of reflected BSDE,is done in Elie and Kharroubi [35]. Another interesting result is the one contained inHamadene and Morlais [45], where the BSDE system is linked to the viscosity solutionof a system of PDE with connected obstacles associated to the switching problem.Extensions to the case of non diffusive noises have also been done. One of the first isHamadene and Hdhiri [48], where a Poisson random measure is added to the two modescase. In Hamadene and Zhao [51] the case where the noises are Brownian motion anda Teugels martingale is studied, again with the help of a system of reflected BSDE.In this chapter we move in this direction, establishing the result in the case of Wienerprocess and marked point process.

6.1 Preliminaries and formulation of the problem

We put ourselves again in the setting described in section 2.2, with a fixed probabilityP, a point process p(dtde) with values in the measurable space (E, E) and a Brownianmotion on RdW . Again Ft is the natural completed filtration generated by p andW . Pdenotes again the predictable σ-algebra. Let the compensator of p be φs(de)dAs, withA continuous. Fix a terminal time T . In the following E(H) denotes the Doleans-Dadeexponential of the process

∫ ·0

∫E

(Hs(e)− 1)q(dsde), that is

E(H)t =∏

0<Tn≤T

HTn(ξn)e∫ t0

∫E(1−Hs(e))φs(de)dAs .

Let J = 1, . . . ,m. Consider now the following optimal switching problem.For each mode i ∈ J , a terminal reward ξi is given, as well as running gains f is and

gis. Moreover, for each i ∈ J , we consider a non-negative process ρis(e) that is P ⊗ Emeasurable. The cost of switching from mode i to mode j at time t is given by Ct(i, j),a non-negative process.

As always, the controller can switch between modes by choosing switching timesand actions. In particular, we define for each mode i the sets Ai of possible switchingdestinations as Ai = J \ i. A strategy is a sequence of couples (θn, αn) where θn isa stopping time and αn is a J -valued Fθn-adapted random variable. The law of thepoint process depends on the switching status, as the switching mode sets a differentcompensator for the point process through the functions ρi. We assume the followingon the ρi and A

Assumption (6-S): The ρi satisfy assumption (2-K) for some M and A satisfies as-sumption (2-K’) for some η > 3 +M4, that is 0 ≤ ρt(e) ≤M and E

[eηAT

]<∞.

Define M ′ = max(|M − 1|, 1). In the following we will often use an absolutelycontinuous change of probability “a la Girsanov”, where the “Girsanov kernel” will beone of the ρi or a combination of them, as described in section 2.1.2. Let us first definean optimal switching strategy and the corresponding switched kernel.

Definition 6.1. A strategy is called admissible if P (θn < T ∀n) = 0. The set ofadmissible strategies such that (θ0, α0) = (t, i) is denoted as Ait.

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6.1. Preliminaries and formulation of the problem

For a strategy (θ0, α0) we define the process a indicating the current mode as

as =∑k≥1

αk−11(θk−1,θk](s) (6.1)

When the controller changes mode, the law of the process changes in the followingway. For a strategy a ∈ Ait, consider the following P ⊗ E-measurable process process

ρas =∑k≥1

1(θk−1,θk](s)ραk−1s (e) (6.2)

Now introduce the supermartingale Lat = E(ρa):

Lat =

∏k≥1

ρaTn(ξn)e∫ t0

∫E(1−ρas (e))φs(de)dAs (6.3)

When La is a martingale, we define the absolutely continuous probability Pa P as

Pa

P= La

T .

In this case, under Pa the compensator of p becomes

νa(dsde) = ρasφs(de)dAs. (6.4)

Remark 6.2. We let the ρi to attain the value zero. This allows us to treat the casein which we have modes in which the compensator becomes zero at times, and thusthe probability of having jumps is null. On the other hand, it introduces some technicaldifficulties since the induced probabilities are not necessarily equivalent to the referenceone. To overcome this, we need to use some approximation techniques as we will seein the following.

Thanks to assumptions above, we have that Lat is a martingale for any choice of

a ∈ Ait:

Proposition 6.3: Let a ∈ Ait be fixed. Under assumption (6-S), La is a P-martingalewith supt∈[0,T ] E [(La

t )2] < +∞. If H is a P ⊗ E-measurable process such that

E[∫ T

0

∫E

eβAs|Hs(e)|2φs(de)dAs]

for some β > 0, then∫ t

0

∫EHs(e)q

a(dsde) is a Pa-martingale, where qa(dsde) =p(dsde)− ρs(e)aφs(de)dAs. Also if M is a P-martingale of the form

Mt = M0 +

∫ t

0

ZsdWs

for some process Z in L2,0(W ), then M is also a Pa-martingale.

Proof. From the definition of ρa we have that ρa verifies assumption (6-S) too, that is0 ≤ ρat (e) ≤M . We are then in the conditions of proposition 2.9

The last assertion is covered by lemma 2.12

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We now formulate the optimal switching problem. Introduce the target quantity tomaximise

J(t, i, a) = Ea

[ξaT +

∫ T

t

fass dAs +

∫ T

t

gass ds−∑k≥1

Cθk(αk−1, αk)

∣∣∣∣∣Ft]. (6.5)

We then have the value function


J(t, i, a). (6.6)

The aim is to find this optimal value and characterize the optimal strategy.In order to tackle the problem, we will represent the value function of each mode

through the use of a system of reflected BSDE. We introduce the following system ofm Backward equations:

Y it = ξi +

∫ Ttf isdAs +

∫ Ttgisds+

∫ Tt

∫EU is(e)(ρ

is(e)− 1)dAs


∫ TtZisdWs +Ki

T −Kit

Y it ≥ max

j∈Ai(Y j

t − Ct(i, j))∫ T0

(Y it −max

j∈Ai(Y j

t − Ct(i, j)))dKit = 0.

(6.7)

In (6.7) we see that the generator of the i-th equation only depends on U i, and itdoes so in a very specific way. This will allow us to consider the system not only underthe reference probability P, but also under the probability Pi induced by the kernel ρi.In this second case the whole term

+

∫ T

t

∫E

U is(e)(ρ

is(e)− 1)dAs −

∫ T

t

U is(e)q(dsde)

will be a Pi martingale. This means we can incorporate in the system of BSDE the factthat in the switching problem the compensator in mode i changes to ρis(e)φs(de)dAs.

First we start with some assumptions, that will serve both to assure that the switchingproblem is well posed and that there exists a solution to the system of BSDE:

Assumption (6-D): For i ∈ J and for some β > (M ′)2

i) ξi is a given FT measurable variable such that Ei[eβAT |ξi|2] <∞.

ii) f is is a progressive process in L2,β(A).

iii) gis is a progressive process in L2,β(W ).

iv) Ct(i, j) for i, j ∈ J are continuous adapted processes. Ct(i, j) ≥ 0 for i 6= j andCt(i, i) = 0 and

inft

(Ct(i, j) + Ct(j, l)− Ct(i, l)) > 0 for all i 6= j 6= l. (6.8)

Moreover, for all i, j ∈ J , E[supt e

βAtCt(i, j)]<∞.

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6.2. Existence of a solution to the system of RBSDE

v) It holds that for all i, j ∈ J ξi ≥ ξj − CT (i, j) a.s.

We recall that the L2,β spaces have been defined in section 2.2. Assumptions i),ii), iii) and v) are the usual ones needed for the reflected BSDE, while iv) is typical ofswitching problems. In particular (6.8) means that it is not possible to switch from i toj by switching to k in the middle for a lower cost.Remark 6.4. The condition (6.8) is a bit stronger than the other usual condition takenin switching problems, that is the no free loop property. This property states that forany cycle of indexes of any length p j1, j2, . . . , jp = j1 it must hold that

p−1∑k=1

Ct(jk, jk+1) > 0.

Details of how (6.8) implies the no free loop property can be found in the proof ofproposition 6.12.

A solution to the BSDE is (Y i, U i, Zi, Ki)i∈J where, for all i ∈ J and for all(M ′)2 < β < β,

• Yi is an adapted cadlag process in L2,β(A) ∩ L2,β(W ).

• Ui is in L2,β(p).

• Zi is in L2,β(W ).

• Kit is an increasing continuous process in I2.

Remark 6.5. We have formulated the problem in the case of a marked point processand a Brownian motion, but it is of course possible to consider the case where only amarked point process is involved. In that case it is enough to put g ≡ 0 and consider alldata adapted to the filtration generated by the point process. The solution would thenbe just (Y i, U i, Ki)i∈J . In some sense, we could say that the Wiener process does notmodify the nature of the current problem. This is because the change of probability forthe point process does not in any way modify the behaviour of the diffusive part.

6.2 Existence of a solution to the system of RBSDE

In this section we obtain existence of a solution to the system (6.7). To this end weconstruct the solution iteratively, where at step n, the barrier is represented by the so-lution at step n − 1. This technique was used for example in Hamadene and Zhang[50], Hamadene and Zhao [51] and Chassagneux, Elie, and Kharroubi [13]. One of themost important tools here is the comparison theorem, as we have to show that our iter-ative scheme converges increasingly to some point and that this point is the solution welook for. The comparison theorem cannot be applied in all cases (for example when wecompare a reflected BSDE with a one without minimal push condition), thus in someparts we resort to direct comparisons. What we will obtain is

Theorem 6.6: Under assumptions (6-S) and (6-D), a solution (Y i, U i, Zi, Ki)i∈J tothe system (6.7) exists. For each i ∈ J , (Y i, U i, Zi, Ki) is in (L2,β(A) ∩ L2,β(W )) ×L2,β(p)× L2,β(W ).

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From now on, let assumptions (6-D) and (6-S) hold. Fix a β such that (M ′)2 < β =

β − δ for some fixed δ > 0. First consider the function defined on [0, T ]× R× E as

hs(u, e) = umaxi∈J

(ρis(e)− 1)1u≥0 + umini∈J

(ρis(e)− 1)1u<0.

Notice that g is Lipschitz on u, indeed

|hs(u′, e)− hs(u, e)| ≤ max(|max

i(ρis(e)− 1)|, |min

i(ρis(e)− 1)|

)|u′ − u|

≤M ′|u′ − u|,

Thus∫Ehs(U(e)) satisfies the special Lipschitz condition:

|∫E

hs(U′(e))− hs(U(e))φs(de)| ≤

∫E

M ′|U ′(e)− U(e)|φs(de)

≤M ′(∫

E

|U ′(e)− U(e)|2φs(de))1/2

.

Define for i ∈ J (Y i,0, U i,0, Zi,0) as solution to

Y i,0t = ξi +

∫ T

t

f isdAs +

∫ T

t

gisds+

∫ T

t

∫E

U i,0s (e)(ρis(e)− 1)φs(de)dAs

−∫ T

t

∫E

U i,0(e)q(dsde)−∫ T

t

Zi,0s dWs, (6.9)

and (Y , U , Z) solution to

Yt = maxi|ξi|+

∫ T

t

maxi|f is|dAs +

∫ T

t

maxi|gis|ds

+

∫ T

t

∫E

hs(Us(e), e)φs(de)dAs −∫ T

t

∫E

U(e)q(dsde)−∫ T

t

ZsdWs. (6.10)

Proposition 6.7: Let assumptions (6-S) and (6-D) hold. For all i ∈ J , there existunique solutions (Y i,0, U i,0, Zi,0) and (Y , U , Z) in (L2,β(A) ∩ L2,β(W )) × L2,β(p) ×L2,β(W ) to (6.9) and (6.10) respectively. Moreover it holds that Y i,0

t ≤ Yt P-a.s. andthat

E

[supt∈[0,T ]

eβAt(|Y i,0t |2 + |Yt|2

)]<∞. (6.11)

Proof. Existence and uniqueness in (L2,β(A) ∩ L2,β(W )) × L2,β(p) × L2,β(W ) areprovided by theorem 2.14

To prove that Y i,0t ≤ Yt we can use the comparison result 2.15. Indeed it is sufficient

to notice that

f is +

∫E

us(e)(ρis(e)− 1)φs(de) ≤ max

i|f is|+

∫E

hs(us(e), e)φs(de).

This is true thanks to the definition of h. Indeed if we explicit it we have

us(e)(ρis(e)−1)−us(e) max

i∈J(ρs(e)

i−1)1us(e)≥0−us(e) mini∈J

(ρs(e)i−1)1us(e)<0 ≤ 0.

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As for the last point of the proposition, it can be shown directly. Let us see it for Y ,since for Y i,0 is analogous. Take expected value conditioned on Ft on the equation forY , then consider the absolute value:

eβAt/2|Yt| ≤ E[eβAt/2|ξ|+ eβAt/2

∫ T

t

|fs|dAs + eβAt/2∫ T

t

|gs|ds∣∣∣∣ Ft]

+ E[+eβAt/2

∫ T

t

∫E

|hs(Us)(e)|φs(de)dAs∣∣∣∣ Ft]

≤ E

[eβAT /2|ξ|+ 1

β1/2

(∫ T

0

eβAs|fs|2dAs)1/2

∣∣∣∣∣ Ft]

+ E

[√T

(∫ T

0

eβAs|gs|2ds)1/2

∣∣∣∣∣ Ft]

+M ′

β1/2E

[(∫ T

0

∫E

eβAs|Us(e)|2φs(de)dAs)1/2

∣∣∣∣∣ Ft].

(6.12)

The last inequality holds thanks to the following

eβAt/2∫ T

t

∫E

|h(Us(e))|φs(de)dAs

≤M ′eβAt/2∫ T

t

∫E

|Us(e)|φs(de)dAs

= M ′eβAt/2∫ T

t

∫E

e−βAs/2eβAs/2|Us(e)|φs(de)dAs

≤M ′eβAt/2(∫ T

t

e−βAsdAs

)1/2(∫ T

0

∫E


= M ′eβAt/2

(e−βAt − e−βAT

β1/2

)1/2(∫ T

0

∫E


≤ M ′

β1/2

(∫ T

0

∫E


,

and analogously for the part with f . As for the integral of g we have

eβAt/2∫ T

t

|gs|ds ≤∫ T

t

eβAs/2|gs|ds ≤√T

(∫ T

0

eβAs|gs|2dAs)1/2

.

The term on the right of inequality (6.12) is a martingale. By Doob’s inequality, theexpected value of its sup is bounded by the quantity

CE[eβAT |ξ|2 +

1

β

∫ T

0

eβAs|fs|2dAs + T

∫ T

0

eβAs|gs|2ds]

+(M ′)2

βE[∫ T

0

∫E

eβAs|Us(e)|2φs(de)dAs],

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which is finite.

Notice that (Y , U , Z, K), with K ≡ 0, solves for all i the Reflected BSDEYt = ξ +

∫ Tt

maxi |f is|dAs +∫ Tt

maxi |gis|ds+∫ Tt

∫Ehs(Us(e))φs(de)dAs

−∫ TtU(e)q(dsde)−

∫ TtZsdWs + KT − Kt

Yt ≥ maxj

(Yt − Ct(i, j))∫ T0

(Yt −maxj

(Yt − Ct(i, j)))dKt = 0

(6.13)Consider now the sequence of RBSDEs:

Y i,nt = ξi +

∫ Ttf isdAs +

∫ Ttgisds+

∫ Tt

∫EU i,ns (e)(ρis(e)− 1)φs(de)dAs)

−∫ TtU i,ns (e)q(dsde−

∫ TtZisdWs +Ki,n

T −Ki,nt

Y i,nt ≥ max

j∈Ai(Y i,n−1

t − Ct(i, j))∫ T0

(Y i,nt− −max

j∈Ai(Y i,n−1

t− − Ct(i, j)))dKi,nt = 0

(6.14)

Proposition 6.8: For all i ∈ J and for all n ≥ 1, (6.14) admits a unique solution in(L2,β(A) ∩ L2,β(W )) × L2,β(p) × L2,β(W ). Moreover it holds that Y i,0

t ≤ Y i,nt ≤

Y i,n+1t ≤ Yt.

Proof. We have the starting point Y i,0, and existence and uniqueness of Y i,1 is providedby theorem 3.17. For every n, we can construct the solution that has maxi∈Ai(Y

j,n−1−Ct(i, j)) as known barrier. First we show by direct verification that for all i, Y i,1 ≥ Y i,0.By considering the difference Y = Y i,1 − Y i,0 we obtain, noting that Ki,1

T −Ki,0t ≥ 0,

Yt ≥∫ T

t

∫E

Us(e)(ρs(e)−1)φs(de)dAs−∫ T

t

∫E


t

ZsdWs. (6.15)

The right hand term in (6.15) is a Pρi-martingale, but since Pρi is not equivalent to P,we cannot conclude that Yt ≥ 0 P-a.s. by taking expectation. Instead we add to bothsides of the equation the quantity

ε

∫ T

t

∫E

Us(e)φs(de)dAs,

for any ε < ε, where ε is such that 3 + (M + ε)4 < η. We obtain

Yt + ε

∫ T

t

∫E

Us(e)φs(de)dAs ≥ −∫ T

t

∫E

Us(e)(p(dsde)− (ρs(e) + ε)φs(de)dAs)

−∫ T

t

ZsdWs. (6.16)

ρi+ε satisfies assumptions (6-S) and, as in section 2.1.2, it induces a probability Pρi+ε ∼P through the “Girsanov” martingale

Lρi+εt = E(ρi + ε),

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for which the results in 2.9. The term in the right hand side of (6.16) is a Pρi+ε-martingale we can take expected value under Pρ+i conditional on Ft and obtain thatfor all ε < ε

Yt + εEρ+ε

[∫ T

t

∫E

Us(e)φs(de)dAs

∣∣∣∣ Ft] ≥ 0 Pρi+ε-a.s.

and thus also P-a.s. Next we need to show that for ε→ 0, the term

εEρ+ε

[∫ T

t

∫E

Us(e)φs(de)dAs

∣∣∣∣ Ft]→ 0.

We only need to show that Eρ+ε[∫ T

tUs(e)φs(de)dAs

∣∣∣ Ft] → 0 is bounded by a finiterandom variable. We have

Eρ+ε

[∫ T

t

Us(e)φs(de)dAs

∣∣∣∣ Ft] ≤ Eρ+ε

[∫ T

t

∫E

|Us(e)|φs(de)dAs∣∣∣∣ Ft]

=E[Lρ

i+εT

∫ Tt


∣∣∣ Ft]Lρ

i+εt

≤ E

[(Lρ

i+εT

Lρi+εt

)∣∣∣∣∣ Ft]

· E

[(∫ T

t

∫E

|Us(e)|φs(de)dAs)2∣∣∣∣∣ Ft

]≤ 1

βE[eηAT

∣∣ Ft]· E[∫ T

t

eβAs|Us(e)|2φs(de)dAs∣∣∣∣ Ft] ,

where in the last line we used relation (2.8) and the usual trick to estimate the square ofan integral with respect to the compensator.

Using the comparison theorem 4.4, we also obtain that Y ≥ Y i,1 for all i. In-deed, the obstacle maxj(Y

j,0 − Ct(i, j)) for Y i,1 is smaller than the one for Y , andwe can transform the equation for Y into the suitable form by adding and subtract-ing

∫ Tt

∫EUs(e)(ρ

is(e) − 1)φs(de)dAs. This way the generator of the equation for Y

becomes

maxi|f is|+

∫E

[hs(Us(e), e)− Us(e)(ρis(e)− 1)

]≥0

φs(de) +

∫E

Us(e)(ρis(e)− 1)φs(de)

and thus

f is ≤ maxi|f is|+

∫E

hs(Us(e), e)− Us(e)(ρis(e)− 1)φs(de),

and we can apply the theorem. By induction we can then prove that Y i,n ≤ Y i,n+1 ≤ Y ,repeating the same reasoning.

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This means that we have an increasing sequence of processes Y i,n, and then thereexists a limit Y i such that

Y i,nt Y i

t ≤ Yt,

for all i, for all t, P-a.s. By dominated convergence theorem we also have that ‖Y i,n −Y i‖L2,β(A) → 0 as n→∞.We have, as in proposition 3.6, the following bound for the norms.

‖Y i,n‖L2,β(A) + ‖Y i,n‖L2,β(W ) + ‖U i,n‖L2,β(p) + ‖Zi,n‖L2,β(W ) ≤ E[eβAT |ξ|2

]+ E

[∫ T

0

eβAs|gis|2ds]

+ E[∫ T

0

eβAs|f is|2dAs]

+ E

[supt∈[0,T ]

eβAt|Y i,nt |2

]

+ E

[supt∈[0,T ]

e(β+δ)At |maxj∈Ai

(Y j,nt − Ct(i, j))|2

]. (6.17)

The only difference here is the added term∫EU i,ns (ρis(e) − 1)φs(de), but since β >

(M ′)2 it gets absorbed easily. Now since Y i,n is a nondecreasing sequence, it holds that

|Y i,nt | ≤ max|Y i,0

t |, |Yt|.

Taking into account proposition 6.7, we have that

E

[supt∈[0,T ]

eβAt max|Y i,0t |, |Yt|

]and thus the last term in (6.7) is bounded by a constant independent of n.

The sequence (U i,n, Zi,n) is bounded in norm thanks to (6.17). Then we can applyproposition 2.16 and obtain the existence of (U i, Zi, Ki) such that (Y i,nk , U i,nk , Zi,nk)

w

(Y i, U i, Zi) in the Hilbert space (L2,β(A)∩L2,β(W ))×L2,β(p)×L2,β(W ), for a suit-able subsequence nk, and the following holds:

Y it = ξi +

∫ Ttf isdAs +

∫ Ttgisds+

∫ Tt

∫EU is(e)(ρ

is(e)− 1)φs(de)dAs

−∫ TtU i(e)q(dsde)−

∫ TtZisdWs +Ki

T −Kit

Y it ≥ max

j∈Ai(Y j

t − Ct(i, j)(6.18)

Now consider, for i ∈ J , the RBSDE with known obstacle

Y it = ξi +

∫ Ttf isdAs +

∫ Ttgisds+

∫ TtU is(e)(ρ

is(e)− 1)φs(de)dAs


∫ TtZisdWs + Ki

T − Kit

Y it ≥ max

j∈Ai(Y j

t − Ct(i, j))∫ T0

(Y it −max

j∈Ai(Y j

t − Ct(i, j)))dKit = 0.

(6.19)

The solution (Y , U , Z, K) exists thanks to theorem 3.17. Thanks to the comparisonresult 4.4, we have that Y i,n

t ≤ Y it for all n, and thus

Y it ≤ Y i

t . (6.20)

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We want to prove the reverse inequality, which is a bit more involved. To this end wewill use a couple of lemmas. Fix ε such that 3+(M+ ε)4 < η. Here we consider again,for all i ∈ J and for all ε < ε, the probability Pρi+ε introduced as

Pρi+ε

dP= E(ρi + ε)

For each i ∈ J define

ηit = ξi1t≥T +

∫ t∧T

0

f isdAs +

∫ t∧T

0

gisds+ maxj∈Ai

(Y jt − Ct(i, j))1t<T. (6.21)

Lemma 6.9:

Y it +

∫ t

0

f isdAs +

∫ t

0

gisds+ ε ess supτ≥t

Eρi+ε[∫ τ

t

∫E

U is(e)φs(de)dAs

∣∣∣∣Ft] (6.22)

is the Pρi+ε Snell’s envelope of ηi.

Proof. This is done like the proof of proposition 3.12. By adding ε∫ τt

∫EU is(e)φs(de)dAs

to both sides of the equation solved by Y i, considered between t and a generic τ ≥ twe obtain

Y it + ε

∫ τ

t

∫E

U is(e)φs(de)dAs = Y i

τ +

∫ τ

t

f isdAs +

∫ τ

t

gisds

+

∫ τ

t

U is(e)(ρ

is(e) + ε− 1)φs(de)dAs−

∫ τ

t

U is(e)q(dsde)−

∫ τ

t

ZisdWs + Ki

τ − Kit .

Since K is increasing, and Y iτ ≥ ξi1τ≥T + max

j∈Ai(Y j

τ − Cτ (i, j))1τ<T, we have

Y it + ε

∫ τ

t

∫E

U is(e)φs(de)dAs +

∫ t

0

f isdAs +

∫ t

0

gisds ≥

ξi + maxj∈Ai

(Y jτ − Cτ (i, j))1τ<T +

∫ τ

0

f isdAs +

∫ τ

0

gisds

+

∫ τ

t

U is(e)(ρ

is(e) + ε− 1)φs(de)dAs −

∫ τ

t

Zis −

∫ τ

t

U is(e)q(dsde).

Being a Pρi+ε-martingale, by taking expectation under Pρi+ε the term

+

∫ τ

t

U is(e)(ρ

is(e) + ε− 1)φs(de)dAs −

∫ τ

t

U is(e)q(dsde)−

∫ τ

t

ZisdWs

disappears and we obtain

Y it + εEρi+ε

[∫ τ

t

∫E

U is(e)φs(de)dAs

∣∣∣∣ Ft]+

∫ t

0

f isdAs +

∫ t

0

gisds

≥ Eρi+ε[ξi1τ≥T + max

j∈Ai(Y j

τ − Cτ (i, j))1τ<T +

∫ τ

0

f isdAs +

∫ τ

0

gisds

∣∣∣∣ Ft] .87

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By taking the ess supa∈Ait over τ ≥ t we obtain

Y it +ε ess sup

τ≥tEρi+ε

[∫ τ

t

∫E

U is(e)φs(de)dAs

∣∣∣∣ Ft]+

∫ t

0

f isdAs+

∫ t

0

gisds ≥ Rρi+ε(ηi),

(6.23)where Rρi+ε(ηi) indicates the Snell envelope of ηi under the probability Pρi+ε. For thereverse inequality, consider the stopping time

Dδt = inf

s ≥ t : Ys ≤ max

j∈Ai(Y j

s − Cs(i, j)) + δ

∧ T. (6.24)

We repeat the reasoning above, but in this case KDδt= Kt and

YDδt ≤ ξi1Dδt≥T + maxj∈Ai

(Y j

Dδt− CDδt (i, j))1Dδt<T,

and we obtain

Y it + εEρi+ε

[∫ Dδt

t

∫E

U is(e)φs(de)dAs

∣∣∣∣∣ Ft]

+

∫ t

0

f isdAs +

∫ t

0

gisds

≤ Eρi+ε[ξi1Dδt≥T + max

j∈Ai(Y j

Dδt− CDδt (i, j)) +

∫ Dδt

0

f isdAs +

∫ Dδt

0

gisds

∣∣∣∣∣ Ft]

+δ.

By taking the ess sup and sending δ to zero we obtain

Y it + ε ess sup

τ≥tEρi+ε

[∫ τ

t

∫E

U is(e)φs(de)dAs

∣∣∣∣ Ft]+

∫ t

0

f isdAs

∫ t

0

gisds ≤ Rρi+ε(ηi),

that together with (6.23) completes the proof.

On the other hand we have

Lemma 6.10: The process

Y it +

∫ t

0

f isdAs +

∫ t

0


Eρi+ε[∫ τ

t

∫E

U is(e)φs(de)dAs

∣∣∣∣Ft]is a Pρi+ε-supermartingale such that

Y it +

∫ t

0

f isdAs +

∫ t

0


Eρi+ε[∫ τ

t

∫E

U is(e)φs(de)dAs

∣∣∣∣Ft] ≥ ηit

Proof. Clearly we have

Y it +

∫ t

0

f isdAs +

∫ t

0

gisds ≥

ξi1t≥T + maxj∈Ai

(Y jt − Ct(i, j))1t<T +

∫ t

0

f isdAs +

∫ t

0

gisds = ηit,

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and since ε ess supτ≥t Eρi+ε[∫ τt


∣∣Ft] ≥ 0 we obtain the secondproperty, that is

Y it +

∫ t

0

f isdAs +

∫ t

0


Eρi+ε[∫ τ

t

∫E

U is(e)φs(de)dAs

∣∣∣∣Ft] ≥ ηit

To show it is a supermartingale consider the equation solved by Y i between t anda generic stopping time τ , where we add to both sides of the equation the quantityεEρi+ε

[∫ τt


∣∣Ft] and take expected value under the measure Pρi+ε:

Y it +

∫ t

0

f isdAs +

∫ t

0

gisds+ εEρi+ε[∫ τ

t

∫E

U is(e)φs(de)dAs

∣∣∣∣Ft]= Eρi+ε

[Yτ +

∫ τ

0

f isdAs +

∫ τ

0

gisds+Kiτ

∣∣∣∣ Ft]−Kit .

By taking ess supτ≥t on both sides it becomes

Y it +

∫ t

0

f isdAs +

∫ t

0


Eρi+ε[∫ τ

t

∫E

U is(e)φs(de)dAs

∣∣∣∣Ft]= ess sup

τ≥tEρi+ε

[Yτ +

∫ τ

0

f isdAs +

∫ τ

0

gisds+Kiτ

∣∣∣∣ Ft]−Kit .

The term

ess supτ≥t

Eρi+ε[Yτ +

∫ τ

0

f isdAs +Kiτ

∣∣∣∣ Ft]is a Pρi+ε-supermartingale (it is a Snell envelope) from which we are subtracting anincreasing process. The right hand side then a Pρi+ε-supermartingale and so is the lefthand side.

With this two lemmas in hand, we can show that P-a.s. Yt ≤ Yt.

Proposition 6.11: It holds that P-a.s.

Yt ≤ Yt.

Proof. Since the Snell envelope of a process is the smallest super-martingale that dom-inates the process, by the lemmas above we have that Pρi+ε-a.s.

Y it +

∫ t

0

f isdAs +

∫ t

0


Eρi+ε[∫ τ

t

∫E

U is(e)φs(de)dAs

∣∣∣∣Ft]≤ Y i

t +

∫ t

0

f isdAs +

∫ t

0


Eρi+ε[∫ τ

t

∫E

U is(e)φs(de)dAs

∣∣∣∣Ft] .Since Pρi+ε ∼ P, this holds also P-a.s. We must now show that

ε ess supτ≥t

Eρi+ε[∫ τ

t

∫E

U is(e)φs(de)dAs

∣∣∣∣Ft]→ 0

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for ε→ 0, and that the same holds for the integral of U i. Notice that

ε ess supτ≥t

Eρi+ε[∫ τ

t

∫E

U is(e)φs(de)dAs

∣∣∣∣Ft]≤ εEρi+ε

[∫ T

0

∫E

|U is(e)|φs(de)dAs

∣∣∣∣Ft]First observe that

Eρi+ε[∫ T

0

∫E

|U is(e)|φs(de)dAs

∣∣∣∣ Ft] =E[Lρ

i+εT

∫ T0

∫E|U i

s(e)|φs(de)dAs∣∣∣ Ft]

Lρi+εt

and then by Young inequality

E[Lρ

i+εT

∫ T0

∫E|U i

s(e)|φs(de)dAs∣∣∣ Ft]

Lρi+εt

≤E[(Lρ

i+εT )2

∣∣∣ Ft](Lρ

i+εt )2

+

E[(∫ T

0

∫E|U i

s(e)|φs(de)dAs)2∣∣∣∣ Ft]

4. (6.25)

The last term in the right hand side is P-a.s. finite as we have by Cauchy-Schwarzinequality(∫ T

0

∫E

e−βAs/2eβAs/2|U is(e)|φs(de)dAs

)2

≤∫ T

0

e−βAsdAs

∫ T

0

∫E


≤ 1

β

∫ T

0

∫E

eβAs|U is(e)|2φs(de)dAs,

where the last term is in L1(P). As for the first term on the right hand side of (6.25),thanks to lemma 2.9 we know that it is smaller than

E[eηAT

∣∣ Ft] ,that we know to be finite. Then

ε ess supτ≥t

Eρi+ε[∫ τ

t

∫E

U is(e)φs(de)dAs

∣∣∣∣Ft]≤ εE

[eηAT +

1

4β

∫ T

0

∫E


∣∣∣∣ Ft] ,and when we send ε to zero we obtain the convergence to zero of the term on the lefthand side. The same applies to the integral of U i. This tells us that for all t

Yt ≤ Yt P-a.s.

Since both processes are cadlag, this holds P-a.s. for all t.

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By combining this lemma and (6.20), we obtain that Y is indistinguishable from Y .This entails that∫ T

t

∫E

U is(e)(ρ

is(de)− 1)φs(de)dAs−

∫ T

t

U is(e)q

i(dsde)−∫ T

t

ZisdWs +Ki

T −Kit

=

∫ T

t

∫E

U is(e)(ρ

is(de)−1)φs(de)dAs−

∫ T

t

U is(e)q

i(dsde)−∫ T

t

ZisdWs+K

iT−Ki

t .

Then, reasoning as in the proof of uniqueness 6.15, it is possible to show that U = Usince the inaccessible jumps of both sides are the same, Z = Z by considering thepredictable bracket against

∫ t0(Zs − Zs)dWs and thus that Kt = Kt.

(Y i, U i, Zi, Ki)i∈J is almost the solution to system (6.7), in the sense that is solvesY it = ξi +

∫ Ttf isdA

is +∫ Ttgisds+

∫ Tt

∫EU is(e)(ρ

is(de)− 1)φs(de)dAs

−∫ TtU is(e)q

i(dsde)−∫ TtZisdWs +Ki

T −Kit

Y it ≥ max

j∈Ai(Y j

t − Ct(i, j))∫ T0

(Y it− −max

j∈Ai(Y j

t− − Ct(i, j)))dKit = 0.

(6.26)

We only have to show that the Ki is continuous. This also means that Y i are Up-per Semi Continuous in Expectation along stopping times (USCE, see Kobylanski andQuenez [60]) and so is the barrier of each RBSDE in the system. This is a crucial prop-erty for the existence of optimal stopping times. Note this also means that Y i jumps aretotally inaccessible.

Proposition 6.12: The processes Ki are continuous.

Proof. Since (Ki)i∈J are predictable process, their jump times are predictable stoppingtimes. Also all jumps are non-negative since (Ki)i∈J are increasing. Suppose thenthere exists j1 and τ such that ∆Kj1

τ > 0, with τ a predictable stopping time. Since themartingale part has only totally inaccessible jumps, it holds that

∆Y j1τ = −∆Kj1

τ < 0.

Thanks to the Skorohod condition in 6.26, it holds that

Y j1τ− = max

k∈Aj1(Y k

τ− − Ct(j1, k)).

This means that for some index j2,

Y j2τ− − Cτ (j1, j2) = Y j1

τ− ≥ Y j1τ ≥ Y j2

τ − Cτ (j1, j2)

where the first inequality is because ∆Y j1τ < 0. We deduce ∆Y j2

τ < 0 and ∆Kj2τ > 0.

Then againY j2τ− = max

k∈Aj2(Y k

τ− − Ct(j2, k)),

and there is j3 such that

Y j3τ− − Cτ (j2, j3) = Y j2

τ− ≥ Y j2τ ≥ Y j3

τ − Cτ (j2, j3)

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and −∆Y j3τ = ∆Kj3

τ > 0. Since the set of indexes is finite, by iterating this procedureit is possible to find a finite sequence of numbers j1, . . . , jp = j1 such that

Y j1τ− = Y j2

τ− − Cτ (j1, j2), Y j2τ− = Y j3

τ− − Cτ (j2, j3), . . . , Yjp−1

τ− = Yjpτ− − Cτ (jp−1, jp).

This last set of equalities implies

p−1∑k

Cτ (jk, jk+1) = 0. (6.27)

Using condition (6.8), we can write

Cτ (j1, j2) + Cτ (j2, j3) > Cτ (j1, j3)

Cτ (j1, j2) + Cτ (j2, j3) + Cτ (j3, j4) > Cτ (j1, j3) + Cτ (j3, j4) > Cτ (j1, j4)

...p−1∑k=1

Cτ (jk, jk+1) > Cτ (j1, jp) = Cτ (j1, j1) = 0

which contradicts 6.27. Thus Kj1τ = 0 and Ki are continuous.

Since Ki are continuous, there is no need for limits in the Skorohod condition andwe have a solution for (6.7). Note that all the Y i are USCE and have only inaccessiblejumps.

6.3 Verification theorem

Now that we have established existence of a solution to the system, we can use it torepresent the value function. The idea is the following. We “glue together” the solutionsto the system in accordance with strategies: given a strategy inAit, we start with Y i

t andconsider its dynamic between t and the first switching time. Then we switch to anotherY jt according to the strategy. This way we obtain a “switched process” that contains

the rewards minus the cost, plus a P-martingale part and the integral of the U parts ofthe equations against the ρ. This last piece will be altogether a martingale under theprobability induced by the strategy, thus by taking expected value under that measurewe obtain the gain for this strategy. First,

Definition 6.13. Given a strategy a in Ait, the cumulated switching cost Das is defined

asDas =

∑n≥1

Cθn(αn−1, αn)1θn≤t DaT = lim

s→TDas . (6.28)

Das is a cadlag adapted increasing process. The value function is rewritten as


J(t, i, a) = Ea

[ξaT +

∫ T

t

fass dAs +

∫ T

t

gass ds−DaT

∣∣∣∣Ft] . (6.29)

We have the main result

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6.3. Verification theorem

Theorem 6.14: Let (Y i, U i, Zi, Ki)i∈J be the solution to the system (6.7). Then P-a.s.

Y it = ess sup

a∈AitJ(t, i, a) = v(t, i). (6.30)

Moreover, the strategy a∗ = (θ∗n, α∗n) defined as (θ∗0, α

∗0) = (t, i) and

θ∗n = inf

s ≥ θ∗n−1 : Y

α∗n−1s = max

j∈Aα∗n−1

(Y js − Cs(α∗n−1, j))

(6.31)

α∗n = arg maxj∈Aα∗n−1

(Y kθ∗n− Cθ∗n(α∗n−1, j)

)(6.32)

is an optimal strategy.

Proof. Step 1. We first show it in the case where the ρi also satisfy 0 < c ≤ ρit(e) forsome constant c. Consider (Y i, U i, Zi, Ki)i∈J be the solution to the system (6.7). Leta ∈ Ait, and for this a define

KaT = Ki

θ1−Ki

t +∑n≥1

(Kαnθn+1−Kαn

θn

)and

Uar =

∑n≥0

Uαnr 1θn<r≤θn+1 Za

r =∑n≥0

Zαnr 1θn<r≤θn+1

Now rewrite the equation for Y i between t and θ1:

Y it = Y i

θ1+

∫ θ1

t

f isdAs +

∫ θ1

t

gisds+

∫ θ1

t

∫E

U is(e)(ρ

is − 1)φs(de)dAs

−∫ θ1

t

∫E

U is(e)q(dsde)−

∫ θ1

t

ZisdWs +Ki

θ1−Ki

t

≥(Y α1θ1− Cθ1(i, α1)

)1θ1<T + ξα01θ1=T +

∫ θ1

t

fass dAs +

∫ θ1

t

gass ds

+

∫ θ1

t

∫E

Uas (e)(ρas − 1)φs(de)dAs −

∫ θ1

t

∫E

Uas (e)q(dsde)

−∫ θ1

t

Zas dWs +Ki

θ1−Ki

t

= Y α1θ21θ1<T +

∫ θ2

t

fass dAs +

∫ θ2

t

gass ds

+

∫ θ2

t

∫E

Uas (e)(ρas − 1)φs(de)dAs −

∫ θ2

t

∫E

Uas (e)q(dsde)+

+

∫ θ2

t

Zas dWs + (Ki

θ1−Ki

t) + (Kα1θ2−Kα1

θ1)− Cθ1(i, α1)1θ1<T + ξα01θ1=T,

where we used first the barrier condition and then the equation for Y α1θ1

between θ1 and

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θ2. This process can be repeated until we obtain

Y it ≥ ξaT +

∫ T

t

fass dAs +

∫ T

t

gass ds+

∫ T

t

∫E

Uas (ρas − 1)φs(de)dAs

−∫ T

t

∫E

Uas (e)q(dsde)−

∫ T

t

Zas dWs + Ka

T −DaT ,

which ends thanks to the fact that a is an admissible strategy and thus P(θn < T ∀n ≥0) = 0. This can be rewritten as (forgetting about the non-negative Ka

T )

Y it ≥ ξaT +

∫ T

t

fass dAs +

∫ T

t

gass ds−∫ T

t

Zas dWs

−∫ T

t

∫E

Uas (e)(p(dsde)− ρas(e)φs(de)dAs)−Da

T . (6.33)

Now, by taking Ea expectation, we have that

Y it ≥ Ea

[ξaT +

∫ T

t

fass dAs +

∫ T

t

gass ds−DaT

∣∣∣∣ Ft]− Ea

[∫ T

t

∫E

Uas (e)(p(dsde)− ρas(e)φs(de)dAs) +

∫ T

t

Zas dWs

∣∣∣∣ Ft] ,now the term

∫ Tt

∫EUas (e)(p(dsde)−ρas(e)φs(de)dAs)−

∫ TtZas dWs is a Pa-martingale,

as stated by lemma 2.12. Thus we have:

Y it ≥ Ea

[ξaT +

∫ T

t

fass dAs −DaT

∣∣∣∣ Ft] Pa-a.s. (6.34)

To prove that Y i is indeed the ess sup, consider the strategy a∗ defined above. We firstprove that it is indeed an admissible strategy, by showing that P(θ∗n < T ∀n ≥ 0) = 0.Assume, as done in Hamadene and Zhao [51], that this does not hold and P(θ∗n <T ∀n ≥ 0) > 0. By the definition of a∗ this means that

P(Yα∗nθ∗n+1

= Yα∗n+1

θ∗n+1− Cθ∗n+1

(α∗n, α∗n+1), α∗n+1 ∈ Aa∗n , ∀n ≥ 0

)> 0.

SinceJ is finite, there exists a loop i0, i1, . . . , ik, i0 of elements ofJ and a subsequencenq(ω)q≥0 such that

P(Y ilθ∗nq+l

= Yil+1

θ∗nq+l− Cθ∗nq+l (il, il + 1), l = 0, . . . , k; ik+1 = i0 ∀q ≥ 0

)> 0. (6.35)

Consider now θ∗ = lim θ∗n and the set Θ = θ∗n < θ∗, ∀n ≥ 0. Thanks to property 6.8we know that

P(θ∗ < T ∩Θc) = 0.

Indeed on Θc we know that for some n we have θ∗n = θ∗ for all n ≥ n. This wouldmean that we keep switching over a cycle with cost zero at θ∗ (see proposition 6.12 orremark 6.5). Indeed we would have

Y j0θ∗ = Y j1

θ∗ − Cθ∗(j0, j1), Y j1θ∗ = Y j2

θ∗ − Cθ∗(j1, j2), . . . , Y jkθ∗ = Y j0

θ∗ − Cθ∗(jk, j0),

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and this again contradicts the no free loop property. Now this tells us that θ∗ is not atotally inaccessible stopping time (see Dellacherie and Meyer [24, Chapter 4, 79] orHe, Wang, and Yan [52, Chapter 3,3]), since it must hold that P(Θ) > 0. Since thejumps of the Y i are totally inaccessible, this means that there is no jump at θ∗ and thuswe can take the limit in (6.35) obtaining

P(Y ilθ∗ = Y

il+1

θ∗ − Cθ∗(il, il + 1), l = 0, . . . , k; ik+1 = i0 ∀q ≥ 0)> 0.

This can be rewritten as

P

(k∑l=0

Cθ∗(il, il+1) = 0; ik+1 = i0

)> 0

which contradicts the non free loop property (see again remark 6.5).Now that we know that a∗ ∈ Ait , we write as before,

Y it = Y i

θ∗1+

∫ θ∗1

t

fa∗s

s dAs +

∫ θ∗1

t

ga∗ss ds+

∫ θ∗1

t

∫E

Ua∗

s (e)(ρa∗

s − 1)φs(de)dAs

−∫ θ∗1

t

∫E

Ua∗

s (e)q(dsde)−∫ θ∗1

t

Za∗

s dWs

=(Yα∗1θ∗1− Cθ∗1 (i, α∗1)

)1θ∗1<T + ξα

∗01θ∗1=T +

∫ θ∗1

t

fa∗s

s dAs +

∫ θ∗1

t

ga∗ss ds

+

∫ θ∗1

t

∫E

Ua∗

s (e)(ρa∗

s − 1)φs(de)dAs −∫ θ∗1

t

∫E

Ua∗

s (e)q(dsde)−∫ θ∗1

t

Za∗

s dWs,

but with the difference thatKiθ∗1−Ki

t = 0 thanks to the Skorohod condition in (6.7) andto the way a∗ is defined. Repeating this as before, but with equalities, we obtain that

Y it = ξa

∗T +

∫ T

t

fa∗s

s dAs +

∫ T

t

ga∗ss ds−

∫ T

t

Za∗

s dWs

+

∫ T

t

∫E

Ua∗

s (e)(ρa∗

s − 1)φs(de)dAs −∫ T

t

∫E

Ua∗

s (e)q(dsde)−Da∗

T ,

that gets rewritten as

Y it = ξa

∗T +

∫ T

t

fa∗s

s dAs +

∫ T

t

ga∗ss ds−Da∗

T

−∫ T

t

∫E

Ua∗

s (e)(p(dsde)− ρa∗s φs(de)dAs)−∫ T

t

Za∗

s dWs.

By taking Pa∗-expected value conditional on Ft we obtain that

Y it = J(t, i, a∗) Pa∗-a.s. (6.36)

Since we assumed that the ρi are bounded from below by a constant c > 0, then theprobabilities Pa introduced are equivalent to P and relations (6.34) and (6.36) hold alsoP-a.s. In that case we have that Y i

t = v(t, i) P-a.s. by combining the two relations.

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Step 2. The general case where the ρi can touch zero is more complicated, butboth in the cases of a generic a and a∗ we can proceed in the same way. Take a stepbackwards and consider the relation (6.33)

Y it ≥ ξaT +

∫ T

t

fass dAs +

∫ T

t

gass ds−∫ T

t

Zas dWs

−∫ T

t

∫E

Uas (e)(p(dsde)− ρas(e)φs(de)dAs)−Da

T .

Consider now ε such that 3+(M+ε)4 < η. For all 0 < ε < εwe add ε∫ Tt

∫EUas φs(de)dAs

to both sides of the previous relation, obtaining

Y it + ε

∫ T

t

∫E

Uas φs(de)dAs ≥ ξaT +

∫ T

t

fass dAs +

∫ T

t

gass ds−∫ T

t

Zas dWs

−∫ T

t

∫E

Uas (e)(p(dsde)− (ρas(e) + ε)φs(de)dAs)−Da

T .

We denote by Pa+ε the probability induced by the kernel ρa + ε, and we have Pa+ε ∼ P.Then by taking expectation under Pa+ε, we obtain that P-a.s.

Y it + εEa+ε

[∫ T

t

∫E

Uas φs(de)dAs

∣∣∣∣ Ft] ≥ Ea+ε

[ξaT +

∫ T

t

fass dAs −DaT

∣∣∣∣ Ft](6.37)

We have already seen that (see proposition 6.8)

εEa+ε

[∫ T

t

∫E

Uas φs(de)dAs

∣∣∣∣ Ft]→ 0 for ε→ 0. (6.38)

If we show that for ε→ 0

Ea+ε

[ξaT +

∫ T

t

fass dAs −DaT

∣∣∣∣ Ft]→ Ea

[ξaT +

∫ T

t

fass dAs −DaT

∣∣∣∣ Ft] ,then we obtain that P-a.s. for all a ∈ Ait

Y it ≥ J(t, i, a). (6.39)

Denote by X the square integrable random variable X = ξaT +∫ Ttfass dAs−Da

T . Andconsider the difference between the two terms∣∣Ea+ε [X| Ft]− Ea [X| Ft]

∣∣ =

∣∣∣∣E [La+εT

La+εt

X

∣∣∣∣ Ft]− E[LaT

Lat

X

∣∣∣∣ Ft]∣∣∣∣=

∣∣∣∣E [(La+εT

La+εt

− LaT

Lat

)X

∣∣∣∣ Ft]∣∣∣∣≤ E

[(La+εT

La+εt

− LaT

Lat

)2∣∣∣∣∣ Ft

]1/2

E[X2∣∣ Ft]1/2 .

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It is clear that (La+εT

La+εt

− LaT

Lat

)2

→ 0 P-a.s.

so we want to use the dominated convergence theorem to prove that the conditionalexpected value converges to zero. It holds that(

La+εT

La+εt

− LaT

Lat

)2

≤ C

(La+εT

La+εt

)2

+ C

(LaT

Lat

)2

We already know, thanks to lemma 2.9, that the second term is bounded by√E((ρa)4)TE((ρa)4)t

eη2AT

which is integrable

E

[√E((ρa)4)TE((ρa)4)t

eη2AT

]≤ CE

[E((ρa)4)TE((ρa)4)t

]+ CE

[eηAT

]≤ C + CE

[eηAT

].

For the first term we obtain a similar estimate, we have that(La+εT

La+εt

)2

can be rewrittenexplicitly as

∏t<Tn≤T

(ρtn(ξn) + ε)2e2∫ Tt

∫E(1−ρs(e)−ε)φs(de)dAs

≤∏

t<Tn≤T

(ρtn(ξn) + ε)2e2∫ Tt

∫E(1−ρs(e)−ε)φs(de)dAs

≤∏

t<Tn≤T

(ρtn(ξn) + ε)2e12

∫ Tt

∫E(1−(ρs(e)+ε)4)φs(de)dAs

· e−12

∫ Tt

∫E(1−(ρs(e)+ε)4)φs(de)dAse2AT−2At

=

√E((ρa + ε)4)TE((ρa + ε)4)t

e−12

∫ Tt

∫E(1−(ρs(e)+ε)4)φs(de)dAse2AT−2At

≤

√E((ρa + ε)4)TE((ρa + ε)4)t

e(L+ε)4(AT−At)− 12

(AT−At)+2(AT−At) ≤

√E((ρa + ε)4)TE((ρa + ε)4)t

eη2AT .

The last term is integrable (as above), and thus we have found an integrable functionthat is greater than (

La+εT

La+εt

− LaT

Lat

)2

.

Then we can apply the dominated convergence theorem and obtain that

Ea+ε

[ξaT +

∫ T

t

fass dAs −DaT

∣∣∣∣ Ft]→ Ea

[ξaT +

∫ T

t

fass dAs −DaT

∣∣∣∣ Ft] ,97

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which together with (6.38) tells us that

Y it ≥ J(t, i, a) P-a.s.

Using the same computation we can show that

Y it = J(t, i, a∗) P-a.s.

and thus P-a.s.Y it = v(t, i)

The representation of Y i as an essential supremum, together with the fact that it iscadlag, also tells us that it is unique. Thanks to this we can prove uniqueness of thesolution to the system, in a quite straightforward way:

Proposition 6.15: The solution to the system (6.7) is unique.

Proof. Consider two sets of solutions (Y i, U i, Zi, Ki)i∈J and (Y i, U i, Zi, Ki)i∈J . Fromtheorem 6.14 we know that for all i Y i

t and Y it both coincide with the value function

v(t, i). Since both are cadlag this means that Y i = Y i up to indistinguishability. Thisin turn lets us write, by considering the equations for Y i and Y i that∫ T

t

∫E

U is(e)(ρ

is(e)−1)φs(de)dAs−

∫ T

t

∫E

U is(e)q(dsde)−

∫ T

t

ZisdWs+Ki

T −Kit

=

∫ T

t

∫E

U is(e)(ρ

is(e)−1)φs(de)dAs−

∫ T

t

∫E

U is(e)q(dsde)−

∫ T

t

ZisdWs+K

iT−Ki

t .

(6.40)

Since both sides must have the same jumps, this means that for all jump Tn with markξn

U iTn(ξn) = U i

Tn(ξn).

This means that U = U in L1,0(p), indeed

E[∫ T

0

∫E

|Us(e)− U is(e)|φs(de)dAs

]= E

[∫ T

0

∫E

|Us(e)− U is(e)|p(dsde)

]= E

[ ∑0<Tn≤T

|U iTn(ξn)− U i

Tn(ξn)|

]= 0.

Now (6.40) becomes∫ T

t

(Zis − Zi

s)dWs = KiT −Ki

t − KiT + Ki

t .

Consider it between 0 and T . By taking the predictable bracket against∫ T

0(Zi

s−Zis)dWs

on both sides we obtain that, since the Ki and Ki are finite variation processes,∫ T

0

(Zis − Zi

s)2dWs = 0,

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which tells us that also the Z component is unique. This leaves us with only K:

KiT −Ki

t − KiT + Ki

t = 0.

It is easy to see that by setting t = 0 we obtain KiT = Ki

T and from that for anyt ∈ [0, T ] Ki

t = Kit .

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CHAPTER7Final remarks and future work

In the present work we have studied two control problems for marked point processes,and have solved them using reflected backward stochastic differential equations. Themotivation was to generalize the result already available in the case of point processeswith compensator absolutely continuous with respect to the Lebesgue measure on R.On the other hand we must restrict ourselves, for the moment, to processes that arenon-explosive. This is due to lack of technology, in particular in the representationof martingales when there is a finite accumulation point for jumps (see Jacod [54] formore details). As a by product, we also obtain some useful results like the comparisonfor simple BSDE driven by a marked point process and other specific tools that havebeen used in the work.

There are several possible extensions that come to mind, so let us first focus on theoptimal stopping problem. One interesting track to follow would be to generalize allthe results to the case where the point process admits also partially accessible jumps(removing the assumption that A is continuous). There are some results on BSDEdriven by non quasi-left-continuous random measures as we have already discussed inthe introduction to chapter 3, and of particular interest are the works Bandini [1] andBandini [2]. While in our case with A continuous the jumps of the martingale part(totally inaccessible) are separated from the jumps of the predictable process K, thiswould no longer be true in this extension. The difficulty would be to understand if it ispossible to obtain a minimal push condition all the same. We could start by imposingsome conditions on the barrier process, so that the pushing term K is continuous, andfocus on the construction of the solution as in Bandini [1].

When considering only a marked point process, it is possible to take advantage ofthe special structure of processes adapted to the natural filtration (see again Jacod [54]or He, Wang, and Yan [52, Chapter I.5]). Between two jump times Tn and Tn+1, a

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Chapter 7. Final remarks and future work

predictable process X is represented as a “FTn-measurable” process Xn, and stoppingtimes are equal to Tn plus a FTn-measurable random variable. This could allow usto obtain a finer characterization of the optimal stopping rule. The starting point is totransform the reflected BSDE in a system of “deterministic” equations as in Confortola,Fuhrman, and Jacod [20], in order to obtain a representation of the solution Y and thebarrier h. This could mean that the optimal stopping rule is a deterministic function ofthe last jump that happened, and thus it may be possible to know in advance when tostop.

The optimal switching problem presents more possibilities. First it would be naturalto allow the generators f and g to depend on the solution in a general way, and alsoallow the generators of the i-th equation to depend on the Y j . This has been done inother cases like Hu and Tang [53], Hamadene and Zhang [50], Hamadene and Zhao[51], Chassagneux, Elie, and Kharroubi [13] or Hamadene and Morlais [45]. To thisend one has to establish, for our case, both a general monotonic limit theorem and ageneral comparison theorem for reflected BSDE. This in turn could also help obtaininga more structured way to do the direct comparisons in chapter 6. Another extensionwould be to allow the mode to affect the diffusive part of the dynamics, having a for-ward process whose coefficients depend on the dynamics. In the case of only controlleddrift, the approach on Hu and Tang [53] could be used, since a change of measure in-volving the Brownian motion will not touch the point process. Another idea is presentin Elie and Kharroubi [35] for the diffusive case, where the system is used to representthe value function though a concatenation of optimal stopping problems and a “flowproperty” for the solutions to the system of reflected BSDE. In our case, the expec-tations taken with respect to different probabilities make this approach troublesome,unless we are able to establish a relation between the different measures. We obtaineda constrained BSDE linked to the reflected BSDE with known obstacle. It would benatural to make use of randomization techniques in this switching problem, obtaining asingle reflected BSDE with constraints that can represent the value function as in Elieand Kharroubi [34].

Again the two extension we mentioned for the reflected BSDE, case with A notcontinuous and a study of the stopping times in the particular filtration, are of interest inthis case. Starting from a reflected BSDE with a discontinuous A, obtaining a solutionof the system should be straightforward. Nevertheless, it is not clear if it is possibleto establish continuity of the Ki processes, which was important in representing thevalue function, at least under the typical assumptions. The second extension would beagain the characterization of the optimal stopping rule using the special structure of thefiltration in the case of only marked point processes.

Lastly it would be quite interesting to investigate means to approximate this kind ofprocesses by means of simulation techniques. Several results in this theory (see againBremaud [10]) are obtained through the use of an explicit representation of the com-pensator obtained as sum of the conditional probability of the next jump and mark,conditioned on the last jump and mark. This kind of extension remains largely unex-plored, at least in this general case.

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ii

“thesis” — 2018/1/31 — 12:47 — page 103 — #113 ii

ii

ii

Bibliography

[1] E. Bandini. “Existence and uniqueness for BSDEs driven by a general random measure, pos-sibly non quasi-left-continuous”. In: Electron. Commun. Probab. 20 (2015), no. 71, 13. DOI:10.1214/ECP.v20-4348.

[2] E. Bandini. “Optimal control of Piecewise Deterministic Markov Processes: a BSDE representa-tion of the value function”. In: ESAIM: Control, Optimisation and Calculus of Variations (Jan.2017). DOI: 10.1051/cocv/2017006.

[3] E. Bandini and M. Fuhrman. “Constrained BSDEs representation of the value function in optimalcontrol of pure jump Markov processes”. In: Stochastic Process. Appl. 127.5 (2017), pp. 1441–1474. DOI: 10.1016/j.spa.2016.08.005.

[4] G. Barles, R. Buckdahn, and E. Pardoux. “Backward stochastic differential equations and integral-partial differential equations”. In: Stochastics Stochastics Rep. 60.1-2 (1997), pp. 57–83.

[5] D. Becherer. “Bounded solutions to backward SDE’s with jumps for utility optimization andindifference hedging”. In: Ann. Appl. Probab. 16.4 (2006), pp. 2027–2054. DOI: 10.1214/105051606000000475.

[6] A. Bensoussan and J.-L. Lions. Applications des inequations variationnelles en controle stochas-tique. Methodes Mathematiques de l’Informatique, No. 6. Dunod, Paris, 1978, pp. viii+545.

[7] A. Bensoussan and J.-L. Lions. Controle impulsionnel et inequations quasi variationnelles. Vol. 11.Methodes Mathematiques de l’Informatique [Mathematical Methods of Information Science].Gauthier-Villars, Paris, 1982, pp. xv+596.

[8] A. Bensoussan. Stochastic control by functional analysis methods. Vol. 11. Studies in Mathematicsand its Applications. North-Holland Publishing Co., Amsterdam-New York, 1982, pp. xv+410.

[9] K. A. Brekke and B. Øksendal. “Optimal switching in an economic activity under uncertainty”. In:SIAM J. Control Optim. 32.4 (1994), pp. 1021–1036. DOI: 10.1137/S0363012992229835.

[10] P. Bremaud. Point processes and queues. Martingale dynamics, Springer Series in Statistics.Springer-Verlag, New York-Berlin, 1981, pp. xviii+354.

[11] M. J. Brennan and E. S. Schwartz. “Evaluating natural resource investments”. In: Journal of busi-ness (1985), pp. 135–157.

[12] R. Carmona and M. Ludkovski. “Pricing asset scheduling flexibility using optimal switching”. In:Appl. Math. Finance 15.5-6 (2008), pp. 405–447. DOI: 10.1080/13504860802170507.

[13] J.-F. Chassagneux, R. Elie, and I. Kharroubi. “A note on existence and uniqueness for solutionsof multidimensional reflected BSDEs”. In: Electron. Commun. Probab. 16 (2011), pp. 120–128.DOI: 10.1214/ECP.v16-1614.

103

https://doi.org/10.1214/ECP.v20-4348

https://doi.org/10.1051/cocv/2017006

https://doi.org/10.1016/j.spa.2016.08.005

https://doi.org/10.1214/105051606000000475

https://doi.org/10.1214/105051606000000475

https://doi.org/10.1137/S0363012992229835

https://doi.org/10.1080/13504860802170507

https://doi.org/10.1214/ECP.v16-1614

ii

“thesis” — 2018/1/31 — 12:47 — page 104 — #114 ii

ii

ii

Bibliography

[14] S. Choukroun, A. Cosso, and H. Pham. “Reflected BSDEs with nonpositive jumps, and controller-and-stopper games”. In: Stochastic Process. Appl. 125.2 (2015), pp. 597–633. DOI: 10.1016/j.spa.2014.09.015.

[15] S. N. Cohen and R. J. Elliott. “Existence, uniqueness and comparisons for BSDEs in generalspaces”. In: Ann. Probab. 40.5 (2012), pp. 2264–2297. DOI: 10.1214/11-AOP679.

[16] S. N. Cohen and R. J. Elliott. Stochastic calculus and applications. Second. Probability and itsApplications. Springer, Cham, 2015, pp. xxiii+666. DOI: 10.1007/978-1-4939-2867-5.

[17] S. N. Cohen, R. J. Elliott, and C. E. M. Pearce. “A general comparison theorem for backwardstochastic differential equations”. In: Adv. in Appl. Probab. 42.3 (2010), pp. 878–898. DOI: 10.1239/aap/1282924067.

[18] F. Confortola. “Lp solution of backward stochastic differential equations driven by a marked pointprocess”. In: arXiv preprint arXiv:1611.10157 (2016).

[19] F. Confortola and M. Fuhrman. “Backward stochastic differential equations associated to jumpMarkov processes and applications”. In: Stochastic Process. Appl. 124.1 (2014), pp. 289–316.DOI: 10.1016/j.spa.2013.07.010.

[20] F. Confortola, M. Fuhrman, and J. Jacod. “Backward stochastic differential equation driven by amarked point process: an elementary approach with an application to optimal control”. In: Ann.Appl. Probab. 26.3 (2016), pp. 1743–1773. DOI: 10.1214/15-AAP1132.

[21] S. Crepey. Financial modeling. Springer Finance. A backward stochastic differential equationsperspective, Springer Finance Textbooks. Springer, Heidelberg, 2013, pp. xx+459. DOI: 10 .1007/978-3-642-37113-4.

[22] S. Crepey and A. Matoussi. “Reflected and doubly reflected BSDEs with jumps: a priori estimatesand comparison”. In: Ann. Appl. Probab. 18.5 (2008), pp. 2041–2069. DOI: 10.1214/08-AAP517.

[23] C. Dellacherie. Capacites et processus stochastiques. Ergebnisse der Mathematik und ihrer Gren-zgebiete, Band 67. Springer-Verlag, Berlin-New York, 1972, pp. ix+155.

[24] C. Dellacherie and P.-A. Meyer. Probabilites et potentiel. Chapitres I a IV, Edition entierementrefondue, Publications de l’Institut de Mathematique de l’Universite de Strasbourg, No. XV, Ac-tualites Scientifiques et Industrielles, No. 1372. Hermann, Paris, 1975, pp. x+291.

[25] C. Dellacherie and P.-A. Meyer. Probabilites et potentiel. Chapitres V a VIII. Revised. Vol. 1385.Actualites Scientifiques et Industrielles [Current Scientific and Industrial Topics]. Theorie desmartingales. [Martingale theory]. Hermann, Paris, 1980, pp. xviii+476.

[26] B. Djehiche, S. Hamadene, and A. Popier. “A finite horizon optimal multiple switching problem”.In: SIAM J. Control Optim. 48.4 (2009), pp. 2751–2770. DOI: 10.1137/070697641.

[27] N. El Karoui. “Les aspects probabilistes du controle stochastique”. In: Ninth Saint Flour Prob-ability Summer School—1979 (Saint Flour, 1979). Vol. 876. Lecture Notes in Math. Springer,Berlin-New York, 1981, pp. 73–238.

[28] N. El Karoui and S.-J. Huang. “A general result of existence and uniqueness of backward stochas-tic differential equations”. In: Backward stochastic differential equations (Paris, 1995–1996).Vol. 364. Pitman Res. Notes Math. Ser. Longman, Harlow, 1997, pp. 27–36.

[29] N. El Karoui, S. Peng, and M. C. Quenez. “Backward stochastic differential equations in finance”.In: Math. Finance 7.1 (1997), pp. 1–71. DOI: 10.1111/1467-9965.00022.

[30] N. El Karoui et al. “Reflected solutions of backward SDE’s, and related obstacle problems forPDE’s”. In: Ann. Probab. 25.2 (1997), pp. 702–737. DOI: 10.1214/aop/1024404416.

[31] M. El Otmani. “Reflected BSDE driven by a Levy process”. In: J. Theoret. Probab. 22.3 (2009),pp. 601–619. DOI: 10.1007/s10959-009-0229-3.

[32] R. Elie and I. Kharroubi. “Probabilistic representation and approximation for coupled systemsof variational inequalities”. In: Statist. Probab. Lett. 80.17-18 (2010), pp. 1388–1396. DOI: 10.1016/j.spl.2010.05.003.

104



https://doi.org/10.1214/11-AOP679

https://doi.org/10.1007/978-1-4939-2867-5

https://doi.org/10.1239/aap/1282924067

https://doi.org/10.1239/aap/1282924067


https://doi.org/10.1214/15-AAP1132

https://doi.org/10.1007/978-3-642-37113-4

https://doi.org/10.1007/978-3-642-37113-4



https://doi.org/10.1137/070697641

https://doi.org/10.1111/1467-9965.00022

https://doi.org/10.1214/aop/1024404416

https://doi.org/10.1007/s10959-009-0229-3

https://doi.org/10.1016/j.spl.2010.05.003


ii

“thesis” — 2018/1/31 — 12:47 — page 105 — #115 ii

ii

ii

Bibliography

[33] R. Elie and I. Kharroubi. “Adding constraints to BSDEs with jumps: an alternative to multidi-mensional reflections”. In: ESAIM Probab. Stat. 18 (2014), pp. 233–250. DOI: 10.1051/ps/2013036.

[34] R. Elie and I. Kharroubi. “Adding constraints to BSDEs with jumps: an alternative to multidi-mensional reflections”. In: ESAIM Probab. Stat. 18 (2014), pp. 233–250. DOI: 10.1051/ps/2013036.

[35] R. Elie and I. Kharroubi. “BSDE representations for optimal switching problems with controlledvolatility”. In: Stoch. Dyn. 14.3 (2014), pp. 1450003, 15. DOI: 10.1142/S0219493714500038.

[36] E. H. Essaky. “Reflected backward stochastic differential equation with jumps and RCLL obsta-cle”. In: Bull. Sci. Math. 132.8 (2008), pp. 690–710. DOI: 10.1016/j.bulsci.2008.03.005.

[37] E. H. Essaky and M. Hassani. “General existence results for reflected BSDE and BSDE”. In: Bull.Sci. Math. 135.5 (2011), pp. 442–466. DOI: 10.1016/j.bulsci.2011.04.003.

[38] W. H. Fleming and H. M. Soner. Controlled Markov processes and viscosity solutions. Second.Vol. 25. Stochastic Modelling and Applied Probability. Springer, New York, 2006, pp. xviii+429.

[39] N. Foresta. “Optimal stopping of marked point processes and reflected backward stochastic dif-ferential equations”. In: arXiv preprint arXiv:1709.09635 (2017).

[40] N. Foresta. “Optimal switching problem for marked point process and systems of reflected BSDE”.In: arXiv preprint arXiv:1710.08506 (2017).

[41] M. Fuhrman, H. Pham, and F. Zeni. “Representation of non-Markovian optimal stopping problemsby constrained BSDEs with a single jump”. In: Electron. Commun. Probab. 21 (2016), Paper No.3, 7. DOI: 10.1214/16-ECP4123.

[42] M. Grigorova et al. “Optimal stopping with f-expectations: the irregular case”. In: arXiv preprintarXiv:1611.09179 (2016).

[43] M. Grigorova et al. “Reflected BSDEs when the obstacle is not right-continuous and optimal stop-ping”. In: The Annals of Applied Probability: an official journal of the institute of mathematicalstatistics (2017).

[44] S. Hamadene. “Reflected BSDE’s with discontinuous barrier and application”. In: Stoch. Stoch.Rep. 74.3-4 (2002), pp. 571–596. DOI: 10.1080/1045112021000036545.

[45] S. Hamadene and M. A. Morlais. “Viscosity solutions of systems of PDEs with interconnectedobstacles and switching problem”. In: Appl. Math. Optim. 67.2 (2013), pp. 163–196. DOI: 10.1007/s00245-012-9184-y.

[46] S. Hamadene and Y. Ouknine. “Reflected backward stochastic differential equation with jumpsand random obstacle”. In: Electron. J. Probab. 8 (2003), no. 2, 20. DOI: 10.1214/EJP.v8-124.

[47] S. Hamadene and Y. Ouknine. “Reflected backward SDEs with general jumps”. In: Theory Probab.Appl. 60.2 (2016), pp. 263–280. DOI: 10.1137/S0040585X97T987648.

[48] S. Hamadene and I. Hdhiri. “The stopping and starting problem in the model with jumps”. In: 7(Dec. 2007), pp. 1081803–1081804.

[49] S. Hamadene and M. Jeanblanc. “On the starting and stopping problem: application in reversibleinvestments”. In: Math. Oper. Res. 32.1 (2007), pp. 182–192. DOI: 10.1287/moor.1060.0228.

[50] S. Hamadene and J. Zhang. “Switching problem and related system of reflected backward SDEs”.In: Stochastic Process. Appl. 120.4 (2010), pp. 403–426. DOI: 10.1016/j.spa.2010.01.003.

[51] S. Hamadene and X. Zhao. “Systems of integro-PDEs with interconnected obstacles and multi-modes switching problem driven by Levy process”. In: NoDEA Nonlinear Differential EquationsAppl. 22.6 (2015), pp. 1607–1660. DOI: 10.1007/s00030-015-0338-x.

105

https://doi.org/10.1051/ps/2013036

https://doi.org/10.1051/ps/2013036

https://doi.org/10.1051/ps/2013036

https://doi.org/10.1051/ps/2013036

https://doi.org/10.1142/S0219493714500038

https://doi.org/10.1016/j.bulsci.2008.03.005



https://doi.org/10.1214/16-ECP4123

https://doi.org/10.1080/1045112021000036545

https://doi.org/10.1007/s00245-012-9184-y

https://doi.org/10.1007/s00245-012-9184-y

https://doi.org/10.1214/EJP.v8-124


https://doi.org/10.1137/S0040585X97T987648

https://doi.org/10.1287/moor.1060.0228

https://doi.org/10.1287/moor.1060.0228



https://doi.org/10.1007/s00030-015-0338-x

ii

“thesis” — 2018/1/31 — 12:47 — page 106 — #116 ii

ii

ii

Bibliography

[52] S. W. He, J. G. Wang, and J. A. Yan. Semimartingale theory and stochastic calculus. KexueChubanshe (Science Press), Beijing; CRC Press, Boca Raton, FL, 1992, pp. xiv+546.

[53] Y. Hu and S. Tang. “Multi-dimensional BSDE with oblique reflection and optimal switching”. In:Probab. Theory Related Fields 147.1-2 (2010), pp. 89–121. DOI: 10.1007/s00440-009-0202-1.

[54] J. Jacod. “Multivariate point processes: predictable projection, Radon-Nikodym derivatives, rep-resentation of martingales”. In: Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 31 (1974/75),pp. 235–253. DOI: 10.1007/BF00536010.

[55] I. Karatzas and S. E. Shreve. Methods of mathematical finance. Vol. 39. Applications of Mathe-matics (New York). Springer-Verlag, New York, 1998, pp. xvi+407. DOI: 10.1007/b98840.

[56] I. Kharroubi, N. Langrene, and H. Pham. “A numerical algorithm for fully nonlinear HJB equa-tions: an approach by control randomization”. In: Monte Carlo Methods Appl. 20.2 (2014), pp. 145–165. DOI: 10.1515/mcma-2013-0024.

[57] I. Kharroubi, N. Langrene, and H. Pham. “Discrete time approximation of fully nonlinear HJBequations via BSDEs with nonpositive jumps”. In: Ann. Appl. Probab. 25.4 (2015), pp. 2301–2338. DOI: 10.1214/14-AAP1049.

[58] I. Kharroubi and H. Pham. “Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE”.In: Ann. Probab. 43.4 (2015), pp. 1823–1865. DOI: 10.1214/14-AOP920.

[59] I. Kharroubi et al. “Backward SDEs with constrained jumps and quasi-variational inequalities”.In: Ann. Probab. 38.2 (2010), pp. 794–840. DOI: 10.1214/09-AOP496.

[60] M. Kobylanski and M.-C. Quenez. “Optimal stopping time problem in a general framework”. In:Electron. J. Probab. 17 (2012), no. 72, 28. DOI: 10.1214/EJP.v17-2262.

[61] T. Kruse and A. Popier. “BSDEs with monotone generator driven by Brownian and Poisson noisesin a general filtration”. In: Stochastics 88.4 (2016), pp. 491–539. DOI: 10.1080/17442508.2015.1090990.

[62] N. V. Krylov. Controlled diffusion processes. Vol. 14. Applications of Mathematics. Translatedfrom the Russian by A. B. Aries. Springer-Verlag, New York-Berlin, 1980, pp. xii+308.

[63] G. Last and A. Brandt. Marked point processes on the real line. Probability and its Applications(New York). The dynamic approach. Springer-Verlag, New York, 1995, pp. xiv+490.

[64] M. Ludkovski. “Optimal switching with applications to energy tolling agreements”. PhD thesis.Princeton University, 2005.

[65] A. Papapantoleon, D. Possamaı, and A. Saplaouras. “Existence and uniqueness results for BSDEswith jumps: the whole nine yards”. In: arXiv preprint arXiv:1607.04214 (2016).

[66] E. Pardoux and S. G. Peng. “Adapted solution of a backward stochastic differential equation”. In:Systems Control Lett. 14.1 (1990), pp. 55–61. DOI: 10.1016/0167-6911(90)90082-6.

[67] E. Pardoux. “Backward stochastic differential equations and viscosity solutions of systems ofsemilinear parabolic and elliptic PDEs of second order”. In: Stochastic analysis and related topics,VI (Geilo, 1996). Vol. 42. Progr. Probab. Birkhauser Boston, Boston, MA, 1998, pp. 79–127.

[68] E. Pardoux and A. Rascanu. Stochastic differential equations, backward SDEs, partial differen-tial equations. Vol. 69. Stochastic Modelling and Applied Probability. Springer, Cham, 2014,pp. xviii+667. DOI: 10.1007/978-3-319-05714-9.

[69] S. Peng. “Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type”. In: Probab. Theory Related Fields 113.4 (1999), pp. 473–499. DOI: 10.1007/s004400050214.

[70] S. Peng and M. Xu. “The smallest g-supermartingale and reflected BSDE with single and doubleL2 obstacles”. In: Ann. Inst. H. Poincare Probab. Statist. 41.3 (2005), pp. 605–630. DOI: 10.1016/j.anihpb.2004.12.002.

106

https://doi.org/10.1007/s00440-009-0202-1

https://doi.org/10.1007/s00440-009-0202-1

https://doi.org/10.1007/BF00536010

https://doi.org/10.1007/b98840

https://doi.org/10.1515/mcma-2013-0024





https://doi.org/10.1080/17442508.2015.1090990

https://doi.org/10.1080/17442508.2015.1090990

https://doi.org/10.1016/0167-6911(90)90082-6

https://doi.org/10.1007/978-3-319-05714-9

https://doi.org/10.1007/s004400050214

https://doi.org/10.1007/s004400050214

https://doi.org/10.1016/j.anihpb.2004.12.002

https://doi.org/10.1016/j.anihpb.2004.12.002

ii

“thesis” — 2018/1/31 — 12:47 — page 107 — #117 ii

ii

ii

Bibliography

[71] G. Peskir and A. Shiryaev. Optimal stopping and free-boundary problems. Lectures in Mathemat-ics ETH Zurich. Birkhauser Verlag, Basel, 2006, pp. xxii+500.

[72] H. Pham. Continuous-time stochastic control and optimization with financial applications. Vol. 61.Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2009, pp. xviii+232. DOI:10.1007/978-3-540-89500-8.

[73] M.-C. Quenez and A. Sulem. “BSDEs with jumps, optimization and applications to dynamic riskmeasures”. In: Stochastic Process. Appl. 123.8 (2013), pp. 3328–3357. DOI: 10.1016/j.spa.2013.02.016.

[74] M.-C. Quenez and A. Sulem. “Reflected BSDEs and robust optimal stopping for dynamic riskmeasures with jumps”. In: Stochastic Process. Appl. 124.9 (2014), pp. 3031–3054. DOI: 10.1016/j.spa.2014.04.007.

[75] Y. Ren and M. El Otmani. “Generalized reflected BSDEs driven by a Levy process and an obstacleproblem for PDIEs with a nonlinear Neumann boundary condition”. In: J. Comput. Appl. Math.233.8 (2010), pp. 2027–2043. DOI: 10.1016/j.cam.2009.09.037.

[76] Y. Ren and L. Hu. “Reflected backward stochastic differential equations driven by Levy pro-cesses”. In: Statist. Probab. Lett. 77.15 (2007), pp. 1559–1566. DOI: 10.1016/j.spl.2007.03.036.

[77] M. Royer. “Backward stochastic differential equations with jumps and related non-linear expec-tations”. In: Stochastic Process. Appl. 116.10 (2006), pp. 1358–1376. DOI: 10.1016/j.spa.2006.02.009.

[78] S. J. Tang and X. J. Li. “Necessary conditions for optimal control of stochastic systems withrandom jumps”. In: SIAM J. Control Optim. 32.5 (1994), pp. 1447–1475. DOI: 10.1137 /S0363012992233858.

[79] S. J. Tang and J. M. Yong. “Finite horizon stochastic optimal switching and impulse controls witha viscosity solution approach”. In: Stochastics Stochastics Rep. 45.3-4 (1993), pp. 145–176. DOI:10.1080/17442509308833860.

[80] J. Xia. “Backward stochastic differential equation with random measures”. In: Acta Math. Appl.Sinica (English Ser.) 16.3 (2000), pp. 225–234. DOI: 10.1007/BF02679887.

107

https://doi.org/10.1007/978-3-540-89500-8





https://doi.org/10.1016/j.cam.2009.09.037





https://doi.org/10.1137/S0363012992233858

https://doi.org/10.1137/S0363012992233858

https://doi.org/10.1080/17442509308833860

https://doi.org/10.1007/BF02679887

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