barcelona seminar on stochastic analysis || nonlinear skorohod stochastic differential equations

Nonlinear Skorohod

Stochastic Differential Equations

Rainer Buckdahn

Abstract. Let (T E Ct(R1 ). We provide assumptions on the random variable

G and the process b = (bt(x)) possibly anticipating the driving Wiener process

(Wt ) under which the anticipative stochastic differential equation with Skorohod

integralt t

X t = G + J(T(Xs)dWs + Jbs(W, Xs)ds, t ~ 0,o 0

has a local solution.

1 Introduction

Let (0, F, P) denote the Wiener space, 0 = Co([O, 1)), and (Wt ) be the coordinate process. The theory of the stochastic integration of processes not necessarily adapted to (Wd has been recently developed by several authors, e.g.,Nualart and Pardoux [6]. This theory allows one to study stochastic differentialequations (S.D.E.) whose solution is not adapted, reviews are given in [9] and[4]. A natural class of anticipative S.D.E.s arises when we consider a randominitial value and shift functions which depend on the whole path of (Wd. Equations of this type with Stratonovich integral have been studied, e.g. by Oconeand Pardoux [8], [9], with Skorohod integral by Buckdahn [3], [4].The purpose of this paper is to study the S.D.E.

t t

X t = G +J(T(Xs)dWs +Jbs(W, Xs)ds, t ~ 0, (1.1)o 0

twhere the stochastic integral J(T(Xs)dWs is defined in the Skorohod sense. For

othe functions (Ts(W,x) = O:sX and bs(W,x) = (3sx, (o:d,((3d E £2([0,1)) it hasbeen considered by Shiota [10] and Ustunel [11]. Under rather weak assumptionswhich allow (O:t) and ((3t) to depend on the whole path of (Wt ), this equationhave been studied by Buckdahn [3], [4] using the Girsanov transformation. In [3]

D. Nualart et al. (eds.), Barcelona Seminar on Stochastic Analysis© Birkhäuser Verlag Basel 1993

22 R. Buckdahn

this has been employed in order to establish the uniqueness of a possible solutionof (1.1) for a large class of possibly anticipating (O"t(W, x)), (bt(W, x)), while theexistence of a solution has been proved only for deterministic 0", b E Cl(Rl).In order to fill this gap by this paper, we impose on 0" to belong to Ct(Rl)while (bt(W,x)) is some possibly anticipating process. Of course, as in the case0", bE Cl(R1 ), also here the solution is only a local one, whereas a global solutioncan be expected to exist only under fairly restrictive assumptions.The organization of the paper is as follows: In Section 2 we recall basic notionsof the anticipative stochastic analysis which are needed in order to constructthe solution (Xd of (1.1) then in Section 3. The proof of Proposition 3.2, astatement of more technical character, is placed into the Appendix.

2 Basic notions

Let (0, F, P) be the Wiener space, 0 =Co([O, 1]) endowed with the supremumnorm 11.11, and (Wd the coordinate process on O. By S we denote the densesubset of L2 (0) which contains all smooth random variables of the form

and define the Malliavin derivative DF as the process

This provides an unbounded closable linear operator D from L 2 (0) into L2 ([0, 1] x0). We identify D with its closed extension and denote its domain by JD1 ,2.

Proposition 2.1 The operator (D, JD1,2) is local in the sense that

I{F=o}DF =0,. a.e., for all FE JD1 ,2.

We shall also use the space JDk,2 (k ~ 2) which' is the completion of S w.r.t.the norm

k

11F11k,2 =L IIID£FIL2([0,ljl)112,£=0

and the space JDk,oo of all elements F of JDk,2 with finite norm

k

1lFllk,oo = L IIID£FluX>([O,l}l) 1100'£=0

Nonlinear Skorohod stochastic differential equations 23

Proposition 2.2 (cf. Prop. 2.5 (2J, Prop. 2.13 (1]): For some k 2: 1, letF E JI)k,oo. Then, for any c > 0, there exists a sequence (Fn) C S with

IIF - Fnllk,2 -+ °(n -+ (0) such that, for all °~ £. ~ k,

For p = 2, +00 and k 2: 1 we introduce the spaces 1£k,p = LP([O, 1], JI)k,p) with

norm

IIIKlllk,p = IIIKtllk,pb([o,1j)'By 8 denote the adjoint of the operator D: To each K E 1£1,2 the operator 8associates some 8(K) E L2 (0) such that

1

E[8(K)F]=E[! KtDtFdtJ, forallFEJI)l,2. (2.1)

°This 8(K) is uniquely determined by (2.1) and called the Skorohod integral.Moreover,

118(K)1I2 ~ IIIKI1II,2.

Proposition 2.3 The operator (8,1£1,2) is local, i.e.,

I 1 8(K) = 0, a.e., for all K E 1£1,2.{J IK,12 dt=O}o

If K E JL1,2, then also the family of processes K I[o,tj, °~ t ~ 1, belongs toJL1 ,2. This allows us to define the integral process

t! KsdWs = 8(KI[o,t)), °~ t ~ 1.

°In addition to the spaces introduced above we also need the space Lk ,2(0 x R1 )

of measurable mappings

b : [0, 1] x 0 X R1-+ R1

with

(i) bt(w,.) E Ck(R1 ), (t,w) E [0,1] x 0,

(ii) bt(.,x) E JI)k,2, (t,x) E [0,1] X R 1

and finite norm

24 R. Buckdahn

1

(iii) Iblk,2 = L IIU JIDm::t bt(w,x)112([O,11",)dt(N(0,1)+Oo)(dx))1/2112'm+l::;k R' 0

Let Lk,OO(O x R1) be the subset of all F E Lk,2(O X R1) with

fjlIblk,oo = L ID

maxlbt(w, x)l(m+l,l,oo) < 00,

m+i9

where, for convenience 1.I(n,k,p) is used to denote the norm of the space LP([O, l]n xOx Rk).

In analogy to Proposition 2.2 we have the following:

Proposition 2.4 Let b E Lk,OO(O X R1). Then, for any c > 0, there is asequence (bn) of processes which are such that

(i) bn has the form

N

LFk(x)I[tk,tk+tl(t), 0::::; t::::; 1,k=O

where Fk(X) = fk(Wt" ... ,WtN ,x), fk E Cr(RN+l), 0::::; k ::::; N,

and 0 = to < tl < ... < tN+l = 1, N ~ 1,

(ii) Ib - bnlk,2 ~ 0 (n ~ (0)and

(iii) IDm~bnl(m+l,l,oo) ::::; c + IDm~bl(m+l,l,oo), m + e::::; k, n ~ 1.

Finally we introduce the concept of a transformation.

A mapping T of 0 into itself is called a transformation if it is of the form

•Tw =w +JKs(w)ds, wE 0,

o

(2.2)

where K belongs to L2 ([0, 1] xO). We say that the transformation T is absolutelycontinuous if the induced measure PT is absolutely continuous relative to P, and

T is called invertible if there is a version of T and a transformation A such that

T(Aw) = A(Tw) = w, wE O.

We refer the reader to [2] or [4] for the main properties of absolutely continuous

transformations and the stochastic calculus associated to them. Let us recall

the following essential properties of transformations:


Proposition 2.5 (Lipschitz condition) Let Tl, T2 be absolutely continuoustransformations and F E JI)l,2. Then,

Proposition 2.6 (Chain rule) Let T be a transformation of the form Tw =•

w+ JKs(w)ds, where K E ILl,oo, and let FE JI)l,oo. Suppose that either FE So

or T is absolutely continuous. Then F(Tw) belongs to JI)l,oo and

1

DdF(Tw)] = (DtF)(Tw) + j(DsF)(TW)DtKs(W)dS, a.e.o

The following properties concern the convergence of sequences of transforma

tions.

•Proposition 2.7 Let (Tnw = w + JK:(w)ds) be a sequence of absolutely con-

otinuous transformations such that

(i) the sequence (Kn) is convergent in L2([0, 1] x [2) to some process K, and

(ii) the sequence of densities (Ln = d~J,n ) is uniformly integrable.

•Then the transformation Tw =w + JKs(w)ds is absolutely continuous and the

odensity L of T is the limit of (Ln) in the weak topology a(Ll , Loo ) .

•Proposition 2.8 Let (Tnw = w +JK:(w)ds) be a sequence of absolutely con

otinuous transformations which satisfies (i) and (ii) of Proposition 2.7. Assumethat (Fn) is a sequence of random variables that are uniformly bounded and

•converge to some F in L2([2). Then, with the notation Tw = w + JKs(w)ds,

owe have

F(Tw) = L2([2) - lim Fn(Tnw).n--->oo

3 Main result and proof

The aim of this section is to prove the following main result:

26 R. Buckdahn

Theorem 3.1 Let a E Ct(R1), G E JI)2,oo and b E L3,oo(D. X R1). Then, forany L > 0, there is some T E (0,1) and some X E IL1 ,2 such that

t t

X t = G + J a(Xs)dWs + J bs(W, Xs)ds, 0::; t ::; T, a.e. on (3.1)o 0

{w ED.: Ilwll ::; L}.

We will prove this theorem by constructing such a process X. Let f E C2 (R2 )

be the unique solution of the equationy

f(x, y) = x + J a(f(x, z))dz,

o

Fix any L > 0 and any <p E Cgo(R1 ) with compact support, and suppose that<p(x) = 1 for -L ::; x ::; L. Then, for abbreviation, we introduce the processes

gt(w,x) = [bt(w,f(X,Wt )) - ~a(f(x,wt))a'(f(x,wt))] x (3.2)

x ( :x f ) (f (x, wd, wd -1 . <p(Wt),

'yt(w,x) = a'(f(x,wd)<p(wd, (t,w,x) E [0,1) x D. x Rl,

for which we can state the following:

Proposition 3.2 Under the assumptions of Theorem 3.1 and for some c > 0,there are aTE (0,1), a unique process V and a unique family of absolutelycontinuous transformations Tt (0::; t ::; T) of D. into itself

(i) which satisfy the equation

t

Vi(w) = G(w) + J gs(Tsw, V,(w))ds,o

tl\.

Ttw w + J IS (Tsw, Vs(w))ds, 0 ::; t ::; T, a.e.,o

(3.3)

and for which

(ii) the processes V = (Vi), K = (Kt = It(Tt, Vi)) and M = (Mt = gt(Tt ,Vi))belong to IL2,oo, and

T T 1/2

(J JIDsKtl2dSdt) ::; 1-c, a.e.o 0

(3.4)


The proof of Proposition 3.2 turns out to be relatively long and technical. Inorder to concentrate ourselves to the construction of the process X in this section, we shift the proof of this proposition to the Appendix.

Proposition 3.2 allows us to make use of Theorem 4.9 [2]:

Proposition 3.3 If K E ]£1,2 is bounded and satisfies (3.4), then, for eacht E [0, T], the transformation

til.

Ttw =w +1Ks(w)ds,o

wE 0,

is invertible, and Tt as well as At = Tt-1 induce measures PT , and PA, which

are absolutely continuous relative to the Wiener measure P,

L _ dPA,t - dP

t t

exp { - 1KsdWs - ~1K;dso 0

t s-11DrKsDs[Kr(As)](Ts)drds}.o 0

(3.5)

This includes that K(Ad = (Ks(Ad)0::;s::;1 E ]£2,00, 0 ::; t ::; T, the field ofMalliavin derivatives {Dr[Ks(At )], 0 ::; r, s ::; 1,0 ::; t ::; T} is bounded inLoo(O), the function t 1-+ Dr[Ks(Ad] E L2(0) is continuous, 0 ::; r, s ::; 1, and

(3.6)r r

r r 1/2 (J J IDrK s j2drds)1/2(11IDs[Kr(As)](Ts)fdrds) ::; 0 Or T , a.e.

o 0 1 - (J J IDrKs12drds )1/2

o 0

Now we can give the following stronger formulation of Theorem 3.1:

Proposition 3.4 Under the assumptions of Theorem 3.1, and with the notationUt = Vi 0 At, 0 ::; t ::; T, the process

o::; t ::; T, (3.7)

solves S.D.E. (3.1) a.e. on {w EO: IIwll ::; L}.

The proof of Proposition 3.4 mainly bases on Theorem 6.1 [6], the anticipative

Ito formula. We present a special case we need:

28 R. Buckdahn

Proposition 3.5 Let f E C2(R2) and U be a continuous process with finitevariation belonging to JLl,2 such that

r r

(i) E[J J IDrUs l2drds) < 00,

o 0and

(ii) the mapping s f-+ Dr[Us) E L4(0) is continuous in [0, r], uniformly withrespect to r E [0, r).

Then, for each t E [0, r], we have

(3.8)t t

f(Ut,Wt ) = f(Uo,O) + j :xf(Us,Ws)dUs + j ~f(Us,Ws)dWso 0

t tIjEP j fj2+ "2 8y2 f (Us,Ws)ds + 8y8x f (Us, Ws)Ds[Us)ds.o 0

r

If (tyf(us, Ws)Iro,r] (s)) E JLl,2, then J tyf(Us,Ws)dWs is nothing else but theo

Skorohod integral, otherwise it is the local Skorohod integral.

The application of this anticipative Ito formula makes a deeper study of thetransformation At and the process U involved in definition (3.7) of X necessary.First we characterize the processes K and V under At.For this we apply Proposition 2.6 to Ks(Ad in order to compute first Dr[Ks(AdJ,

t

Dr[Ks(At )) = (DrKs)(Ad - j(DuKs)(AdDr[Ku(Ad)dU, (3.9)o

and then DvDr[Ks(AdJ,

DvDr[Ks(At )) = (DvDrKs)(Ad (3.10)t

- j(DxDrKs)(AdDv[Kx(At))dXo

t

- j(DvDuKs)(At)Dr[Ku(At))dUo

t t

+ j j(DxDuKs)(At)Dv[Kx(Ad)Dr[Ku(Ad)dUdXo 0

Nonlinear Skorohod stochastic differential equations

t

- j(DuKs)(At)DvDr[Ku(AdJdU.o

29

Hence, by virtue of (3.4), an easy estimate of (3.9) and (3.10) shows that both

fields {Dr [Ks(AdJ, 0 ~ r, s ~ 1,0 < t ~ T} and{DvDr[Ks(AdJ,O ~ v,r,s ~ 1, 0 ~ t ~ T} are bounded subsets of LOO(O),i.e., the processes (Ks(Ad)o~s9' 0 ~ t ~ T, are uniformly bounded in ][)2,00.In particular, passing to the limit t --t r in (3.9) and then using DvDr[Ks(Ar)Jin (3.10) we see that the field {Dt[Ks(At )], 0 ~ s ~ 1, 0 ~ t ~ T} is bounded in][)l,oo .

Lemma 3.6 Let F E ][)l,oo. Then the process (F(At))o~t~T belongs to ILl,ooand its Malliavin derivative {Ds[F(At)J : 0 ~ s ~ 1, 0 ~ t ~ T} has a versionfor which the function t I-t Ds[F(At)J E L2(0) is continuous, for all 0 ~ s ~ 1.Moreover, if F E ][)2,00, then there is a real CK only depending on K such that

and

In particular, this includes that (F( At))o~t~T E IL2,00.

Proof: By Proposition 2.6 we have

t

Ds[F(At)J = (DsF)(At) - j(DrF)(At)Ds[Kr(AdJdr, (3.11)o

o~ s ~ 1, 0 ~ t ~ T.

Consequently, (F(Ad)o~t~T E ILl,oo. By virtue of Proposition 3.3 for theL2 (0)-continuity of the function t I-t Ds[F(AdJ it suffices to show that oft I-t (DsF)(Ad. But the continuity of t I-t (DsF)(Ad E L2(0) can be derivedfrom Proposition 2.8. For this we only have to check, whether the transformations

tA.

Atw = w - j Ks(Atw)ds, 0 ~ t ~ T,

o

(3.12)

satisfy the assumptions of Proposition 2.7: While the uniform LP(O)-integrabilityof the densities L t = d:;" 0 ~ t ~ T, for some p < 1, is stated in Lemma 4.3

30 R. Buckdahn

[2], the convergence of the shift processes (Ks(Ad)o~s~l, 0 ~ t ~ T, is a consequence of the relation

r 1

E[j IKr(As) _ Kr(AtWdr] 2

ofor all 0 ~ s, t ~ 1 (cf. (4.19), [2)).

If now F E JD2 ,oo, then differentiating equation (3.11) after putting s = t yields

t

DuDdF(AdJ = (DuDtF)(Ad - j(DsDtF)(AdDu[Ks(AdJdSo

t

- j(DuDrF)(At)DdKr(At)]drot t

+ j j (DsDrF)(At)DdKr(Ad]Du[Ks(Ad]dsdro 0

t

- j (DrF)(AdDuDdKr(Ad]dr,o

o~ u ~ 1, 0 ~ t ~ T, a.e.

Now there are no difficulties anymore to complete the proof.

Lemma 3.7 Let F E JDl,oo. Then the process (F(At))O~t~r is pathwise absolutely continuous with respect to the Lebesgue measure,

(3.13)

Proof: If FE S, then (3.13) follows easily by chain rule for differentiation. If,more general, F E JD1,oo, then use Proposition 2.2 in order to approximate F

by a sequence (Fn ) C S in JD1,2 such that sup IIFnlh,oo < 00. This allows us ton

deduce from relation (3.11),

s

Ds[Fn(As)] = (DsFn)(As) - j(DrFn)(As)Ds[Kr(As)Jdr,

oo~ s ~ T, a.e.


and Proposition 2.8 that the sequence of processes (Ds[pn(As)])o::=;s::=;T> n ~ 1,

converges to (Ds[P(As)])o::=;s::=;r in L2 ([0, 7] x 0). Hence, in the relation

t

pn(Ad = pn - JKs(As)Ds[Fn(As)]ds, 0::; t ::; 7,

o

which is true for all pn E S, we can pass to the limit and obtain

t

P(At ) = P - JKs(As)Ds[P(As])ds, 0::; t ::; 7, a.e.o

This completes the proof.

The preceeding lemmata make it possible to give the following characterization

fo the process U = (Ut = Vi 0 At)o::=;t::=;r:

Lemma 3.8 The process U belongs to IL2 ,00, its Malliavin derivative has aversion such that the junction t f---> DsUt E L2(0) is continuous, jor all 0 ::;s ::; 1. Moreover, U is pathwise absolutely continuous relative to the Lebesguemeasure,

ddt Ut = Mt(Ad - Kt(At)DdUt], 0::; t ::; 7, a.e., (3.14)

and Cf,Yt)O::=;t::=;r E IL1,00.

Proof: Recall that

t

Vi G +JMsds, i.e.,

ot

Ut G(At ) +JMs(Adds, 0::; t ::; 7.o

Hence, Lemma 3.7 provides

:t Ut -Kt(Ad{ DdG(At )] (3.15)

t

+JDdMs(At)]dS} + Mt(Ad, 0::; t ::; 7, a.e.o

32 R. Buckdahn

Clearly, the right-hand side belongs to ILl,oo. Moreover, due to Lemma 3.6 thefunction t f-+ Ds[Ut] E £2(0) is continuous and

t

DtlUt]= DtlG(At)] +JDtlMs(At)]ds.o

Finally note that (3.15) and (3.16) provide (3.14).

(3.16)

dIDsUr - DsUvl ::; III dt Ut 1111,00 ·Ir - vi, 0::; r,v::; T.

Lemma 3.8 permits us to apply the anticipative Ito formula in order to proveProposition 3.4.

Remark: In particular, it follows from Lemma 3.8 that the function t f-+

DsUt E £4(0) is continuous, uniformly with respect to 0 ::; s ::; 1. It evenholds

Proof of Proposition 3.4: From Lemma 3.8 and the Remark to this lemmawe know that the assumptions of Proposition 3.5 are satisfied. Moreover, the

processes (Xt}O:::;t:::;r and (O"(Xt})O:::;t:::;r belong to IL2,2. Since

G,

O"'(Xs)O"(Xs),

88x!(Us,Ws){Ms(As) - Ks(As)Ds[Us]}

{bs(W,Xs) - ~O"'(Xs)O"(Xs)}<p(Ws)

- :x!(Us, Ws)O"'(Xs)<p(Ws)Ds[Us]

and82 8

8x8y!(Us,Ws)Ds[Us]= 8x!(Us,Ws)O"'(Xs)Ds[Us],

formula (3.8) provides

t t

X t = G +JO"(Xs)dWs +Jbs(W, Xs)<p(Ws)dso 0t

+~JO"'(Xs)O"(Xs)(l - <p(Ws))dso


t

+ j :x f(Us ,Ws)(7/(Xs)DslUs](1 - ip(Ws))ds, 0 :::; t :::; T, a.e.

o

Consequently,

t t

X t = G + j (7(Xs)dWs + j bs(X, Ws)ds,o 0

0:::; t:::; T, a.e. on {wEn: Ilwll :::;L}.

This completes the proof.

33

Since the statement of Proposition 3.4 covers also that of Theorem 3.1, the main

result is proved now.

Appendix

Our aim is to prove Proposition 3.2. For this we use all notations introducedin Section 3, where we suppose in a first step that the process b is of the form(2.2) and G E S. Then we can use the Picard iteration in order to derive theexistence of a pathwise unique solution (V, T) = (lit(w), Ttw) of equation (3.3).

Lemma A.I Let G E Sand b be of the form (2.2). Then the processes V =(lit)o:St:S1, K = (Kt = 'Yt(Tt, lit))O:St:Sl and M = (Mt = gt(Tt, lit))o:St:S1 belongto /L2,oo. Moreover, for some real Ca ,,,, depending only on (7 and ip, it holds

t

IDrlitl 2 + j IDrKs l2ds (A.2)

o:::; Ca,,,,(l + IIGIII,oo) exp{Ca ,,,, (1 + IllblllI,oo)}·

Proof: From (3.3) and Proposition 2.5 we derive that

t

Dr lit = (DrG + j(Drgs)(Ts,Vs)dS) +o

(A.3)

34 R. Buckdahn

t s

+ j j(Dugs)(Ts, lfs)DrKududs

o 0

DrKt = (Dr"ld(Tt, Vi) + (~ "It }Tt , Vi)DrVi +t

+ j (Du"ld(Tt , Vi)DrKudu,

oo::; r, t, ::; 1.

Thus,

< 3(IDGI(1,o,oo) + ID91(2,1,oo))2 +t s

+3 j IDglr2,1,oo)(j jDrK ul2du )dS +

o 0

t 2

+3 j I:xgl IDrlfsl2ds,

o (1,1,00)

t s

< 3ID"Ilr2,1,oo) + 3 j ID"Ilr2,1,oo) (j IDrKul2du) dso 0

t 2

+3 JI:x "II IDr Vs 1

2ds, 0 ::; r, t ::; 1.

o (1,1,00)

Hence, taking into account that for some real C~,'P only depending on (J and <p

it holds

Illgllh.oo ::; C~,'P(1 + Illbllh.oo),

Ih1111,00 ::; C~,'P'

we obtain (A.2). Relation (A.l) and the fact that V, K and M belong to 1£2,00are clear and can easily be derived from (A.2) now.

For convenience let us denote the right-hand side of (A.2) by c(b, G), fix anysmall c > 0 and define

. { (l-c?}T = mm 1, c(b,G) .

Then, obviously, (A.2) yields (3.4), i.e., Proposition 3.2 holds for all G E S andall b of the form (2.2).


Let now G E D)2,00 and b E L3,00(0 X R1 ). Due to the Propositions 2.2 and 2.4,

for any 8 > 0 there are sequences (Gn) c S and (bn) of processes of the form(2.2) with the following properties:

IIG - Gn1l2,2 - 0 (n - 00), (A.4)

IIGnllk,oo < 8+ IIGllk,oo, k = 0,1,2, n ~ 1,

and

Illb - bnlll3,2 - 0 (n - 00), (A.5)

IllbnIllk,oo < 8 + Illblllk,oo, k =0,1,2,3, n ~ 1.

Now replace b and G in equation (3.3) by bn and Gn, respectively, denote thesolution of the new equation by (Vn, Tn), and put Kn = (Kf = '"Yt(Tr, Vt)).Clearly,

IKfl ~ Ca,<p, 0 ~ t ~ 1, n ~ 1.

Putting

c(b, G) =Ca,<p(l + IIIGIIIi,oo) exp{Ca,<p(l + Illbllli,oo)}we have

(1-C)2

c(bn,Gn) ~ 1- 2c c(b,G), for all n ~ 1,

if we only choose 8 > 0 small enough. Hence, with the notation

. { (1-2c)2}7=mm 1, c(b,G) ,

we obtain (3.4),

(A.6)

T T 1/2(JJIDr K: 12drdS) ~ 1 - c, a.e., for all n ~ 1. (A.7)

o 0

This makes Proposition 3.3 applicable to all pairs (Gn, bn), i.e., the inversetransformation Af = (Tr) -1 exists for all 0 ~ t ~ 7, n ~ 1, and both transfor

mations Af and Tr are absolutely continuous. We associate gn to bn by (3.2),and the same we do with 9 for b. Then, obviously,

Cg =sup Illgnlll3,oo < 00 and Illg - gn1113,2 - 0 (n - 00). (A.8)n

In particular, we have

36

Moreover, from (A.7) and the relations

R. Buckdahn

t

Dr[K~(A~)] = (DrK~)(Af) - I(DuK~)(A~)Dr[K:(A~)]dU,ot

Dr[~n(A~)] = (Dr ~n)(Af) - I(Du~n)(A~)Dr[K:(A~)]du,oo~ r, t ~ 7,

it follows

Thus, (A.2) (A.4) and (A.5) allow us to conclude that with the notation

f<r(w) = 'Yt(w, ~n(A~w)), (t,w) E [0,7] X 0, n ~ 1,

it holdsT 1/2

O~~;T {lIf<rll oo + II (I IDr[f<rWdr) 1100} < 00.o

Since the process (f<nO~t~T is defined such that

t/\.

Ttnw = w + I f<~(T:w)ds, wE 0, 0 ~ t ~ 7, n ~ 1,

o

we can make use of Lemma 2.29 [4] and the proof of Theorem 2.2.1 [4] in order

to get the information that the set of densities

{dPAn dPTn }

M = L~ = dP" £~ = dP' 0 ~ t ~ 7, n ~ 1

is uniformly integrable. This allows us to derive the following result:

Lemma A.2 Under the above assumptions the sequence (~nk:~l converges in

L2 (0), for all t E [0,7], and the sequence of processes (Knk~l converges in

L2 ([0, 7] x 0).

Proof: Usual estimate of (3.3)

T

E[I~n - ~mI2] ~ 3E[1 Ig:(Tsm

, Vsm

) - g':(Tsm

, v:,m) 12 ds] +o


t s

+3IDgnl~2,1,00)E [J J IK~ - K;n12drdS] +o 0

37

t

E [J IK~ - K;n12 dr]o

+3!:x gnl E[jlVs

n -vsm

I2dS]'

(1,1,00) 0

t s

< 2ID1'I~2'1'00)E[J J IK~ - K;n12drdS] +o 0

2 t

+21 :x 1'1 E [J IVsn- VsmI2dS],(1,1,00) 0

o~ t ~ T, m, n ~ 1,

wE 0,

and application of Gronwall's Lemma provides the wished convergence, if we

take into account that by virtue of the uniform integrability ofM the expressionT

E[J Ig~(T;n,Vsm) - gr;'(T;n, Vsm)12dS]

oT

< 2E[J {lg~(W,O) - gs(W,0)12+o

converges to zero, as n, m tend to infinity.

Denote the limit of (Kn) by K and define the transformationtil.

Ttw = w+ J Kr(w)dr,o

for all 0 ~ t ~ T. As we have shown above the assumptions of Proposition 2.7

are satisfied, i.e., the transformation Tt is absolutely continuous for all t E [0, T].Then Proposition 2.8 allows us to deduce that, with the notation lit = £2(0)_lim ~n, we haven-.oo

t

lIt(w) = G(w) + J gs(Tsw, Vs(w))ds,o

38 R. Buckdahn

tAo

Ttw w +J'Ys(Tsw, Vs(w))ds, 0 ~ t ~ T, a.e.o

Consequently (lit, Tt}o<::;t<::;r is a solution of equation (3.3). On the other hand,from Proposition 2.5 we can conclude that this solution is unique in the classof all pairs (V, T), V E 1£2,00 and T = {Tt,0 ~ t ~ T} family of absolutelycontinuous transformations.

Using (A.3) with bn, vn, gn, Tn and K n instead of b, V, g, T and K, re

spectively, arguments analogous to those in the proof of Lemma A.2 show that(Vnk::1 and (Knk,,:l do not converge only in L2([O,T] x 0) but even in 1£1,2,and they have limits which belong to 1£1,00. Moreover, passing to the limit in(A.7) gives (3.4). Since G E JD2,00, (gn) C L3,00(O X R 1)with sup Illgnlkoo < 00

nand'Y E L3,00(O X R 1), a renewed Malliavin differentiation of v n and Kn allowsus to prove the convergence in 1£2,2 and the finiteness of the 1£2,00-norm of their

limits V and K. This completes the proof of Proposition 3.2.

References

[1] Buckdahn, R: Transformations on the Wiener space and Skorohodtype stochastic differential equations. Seminarbericht 105, Sekt. Math.,Humboldt-Univ. Berlin 1989

[2] Buckdahn, R: Anticipative Girsanov transformations. Probab. TheoryRelat. Fields 89 (1991), 211-238

[3] Buckdahn, R: Skorohod Stochastic Differential Equations of DiffusionType. To appear in Probab. Theory Relat. Fields

[4] Buckdahn, R: Anticipative Girsanov Transformation and StochasticDifferential Equations. Seminarbericht 92-2, Fachbereich Mathematik,Humboldt-Univ. Berlin 1992

[5] Kusuoka, S.: The non-linear transformation of Gaussian measure on Banach space and its absolute continuity (1). J. Fac. Sci. Univ. Tokyo, Sect.IA 29 (1982), 567-597

[6] Nualart, D.; Pardoux, E.: Stochastic calculus with anticipating integrands. Probab. Theory Relat. Fields 78 (1988), 535-582


[7] Nualart, D.; Pardoux, E.: Boundary value problems for stochastic differential equations. Ann. Probab. 19 (1991) 3, 1118-1144

[8] Ocone, D.; Pardoux, E.: A generalized It6-Ventzell formula. Applicationto a class of anticipating stochastic differential equations. Ann. Inst. HenriPoincare, Probab. Stat. 25 (1989), 39-71

[9] Pardoux, E.: Applications of anticipating stochastic calculus to stochasticdifferential equations. Preprint 1989

[10] Shiota, Y: A linear stochastic differential equation containing the extended Ito integral. Math. Rep., Toyama Univ. 9 (1986),43-65

[11] Ustunel, A.S.: Some comments on the filtering of diffusions and the Malliavin calculus. In: Korezlioglu, H.; Ustunel, A.S. (eds.) Proc. Silivri Conf.1986. - Berlin, Heidelberg, New York: Springer 1988, 247-266 (Lect. NotesMath., 1316)

Rainer BuckdahnFB, Mathematik, Humblodt-UniversitatUnter den Linden,60-1086 BERLIN, Germany

barcelona seminar on stochastic analysis || nonlinear skorohod stochastic differential equations

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