A HAUSSMANN-CLARK-OCONE FORMULA FOR FUNCTIONALS OF DIFFUSION PROCESSES WITH LIPSCHITZ COEFFICIENTS

We establish a martingale representation formula for functionals of diffusion processes with Lipschitz coefficients, as stochastic integrals with respect to the Brownian motion.


Introduction
Let be a filtered probability space satisfying the usual conditions on which a Ð ß ß ßTÑ H Y Y > standard Brownian motion is defined; is the natural filtration Y generated by and augmented with -null sets.It is well known [9,13] -square integrable martingale, then there exists an -adapted process satisfying, L I ±L ± .> ∞ Q oeIÒQ Ó L.FÞ such that --' ' If the martingale is given by the conditional expectation of some Brownian functional , where is a Frechet differentiable functional defined on then Clark [4] has given an explicit representation of the process , namely L L oe IÒ ÐÐ>ß "Óß FÑÎ Ó Ð.?ß FÑ > > P P -Y where, denotes the signed measure associated with the Frechet derivative of .He also uses this formula to solve a stochastic nonlinear filtering s PÐFÑ problem.Haussmann [6] has extended this integral formula to functionals of Ito processes with smooth causal coefficients .\oe ,Ð>ß \Ñ.>  Ð>ß \Ñ.where denotes the unique solution of the equation of first variation associated to .FÐ?Ñ \ The proof is based on Cauchy-Maruyama approximation.The last formula has been applied to stochastic control problems [8].Later in [7], the author gave another proof of this result based on Girsanov's theorem.Note also that this formula played a key role in Bismut's version of Malliavin calculus [1].Ocone [12] recovered the Haussmann-Clark formula by using Malliavin calculus techniques, and extended it to the class of functionals which are P weakly -differentiable.See Davis [5] for a proof using potential theory techniques.
L The common assumption in all previous works is the differentiability of the coefficients ,ß \ 5 with respect to the state variable.This assumption allows one to show that admits a derivative with respect to initial condition, at least in the sense.In this paper we drop this P # hypothesis.We suppose that are only Lipschitz continuous in the state variable.We ,ß 5 establish an integral representation formula in which is defined by means of generalized FÐ?Ñ derivatives of the coefficients .This is done in the nondegenerate, as well as in the ,ß 5 degenerate case.The main idea in the proof is to show that even when the coefficients are merely Lipschitz, it is possible to associate to the diffusion process a unique linearized \ B > version defined as the distributional derivative of with respect to the initial condition.In \ B > the case where the matrix is nondegenerate, our proof is carried out by using 5 5 Ð>ß BÑ Ð>ß BÑ ‡ an approximation procedure and Krylov's inequality.Roughly speaking, this inequality says that the law of the random variable is absolutely continuous with respect to Lebesgue \ B > measure.This property allows us to define a unique linearized version of the stochastic differential equation (2.3).That is, if we choose two versions of the generalized derivatives of and , then the corresponding transition matrices are equal.
, 5 The method performed in the previous case is no longer valid in the degenerate case, and the sort of derivative (with respect to the initial condition) defined will have no sense.In this case, we make use of a result of Bouleau and Hirsch [2,3] to define a generalized derivative of the process defined on an enlarged probability space.The idea is to consider a slightly different stochastic differential equation defined on an enlarged probability space, where the initial condition will be taken as a random element.This allows us to perform B operations outside negligible sets (in , which are not possible for the initial equation.The BÑ method is inspired from result [2,3], where the authors proved an absolute continuity result, extending the well known Malliavin calculus methods.

Assumptions and Preliminaries
Let be a probability space on which is defined a -dimensional Brownian motion , and let be the -algebra generated by completed with -null sets of Y 5 Y V ' .Let be the space of -valued continuous functions defined on , equipped with the topology of uniform convergence.denotes its Borel -field.
is continuous and grows polynomially.
of bounded variation such that, The assumptions (2.4) and (2.5) guarantee the existence and uniqueness of a strong solution for equation (2.3) . .
processes such that: Let be an Ito process.Then for every Borel function with support in the following inequality holds: where is a constant and stands for the ball of center and radius .

The Nondegenerate Case
Assume that the hypothesis in the last section hold and suppose that the diffusion matrix 5Ð>ß BÑ satisfies the uniform ellipticity condition: Let us denote by the solution of the following first variation equation associated The main result of this section is the following: Theorem 3.1: Under assumptions , and , it holds that 5 .Then by Gronwall's inequality, we have oe OÖM  M ×Þ " # Since for , sup , we have lim sup Therefore without loss of generality, we may suppose that have compact , ß , ß ß . By applying Krylov's inequality (thanks to condition (3.1)), we The fact that can be proved using similar arguments. .Then the Proof: Using the Burkholder-Davis-Gundy, Schwarz and Gronwall inequalities, we obtain Since the coefficients in the linear stochastic differential equation (3.6) are bounded, F 8 : Ð=ß >Ñ >Ñ P : " is uniformly (in bounded in for each .Therefore, the first term in the right hand side is finite.To derive the desired result, it is sufficient to prove the following: where .4 oe "ß #ß á ß .Let us prove the first one.Let be a fixed integer, then it holds that: Let be a continuous function such that if and .
Then for , we have Applying Krlov's inequality we obtain

The Degenerate Case
Throughout this section, the reference probability space is the usual Wiener space.We assume that the coefficients and the functional satisfy the hypotheses of the last section ,ß P 5 except condition (3.1) on the uniform ellipticity of .5 5 Ð>ß BÑ Ð>ß BÑ ‡ Let us recall some preliminaries and notations which will be applied in this section to establish the Haussmann-Clark representation formula.See [2,3] for details and proofs.
Let be the space of continuous functions such that , endowed with the topology of uniform convergence on compact subsets of Equipped with the norm H is a Hilbert space, which is a classical Dirichlet space [2,3] Let us define the Hilbert space which is a general Dirichlet space see [3]) Borel measurable such that -a.e. and is locally absolutely continuous.

Let
be the process defined on the enlarged space , which is Since the coefficients are Lipschitz continuous and grow at most linearly, (4.1) has a ,ß 5 unique, continuous, -adapted solution.Note that equations (2.3) and (4.1) are almost the Y µ > same except that uniqueness for (4.1) is slightly weaker.One can easily prove that the uniqueness implies that for each , , -a.s.
for -almost every where denotes the derivative in the sense of distributions.from the dominate convergence theorem.Using the same technique, we prove that for each 4 oe "ß á ß ., Then as 5 Ä  ∞  In particular, the absolute continuity assumption is satisfied in the one Remark 4.8: dimensional case and the coefficients are time independent such that for some ,ß ÐBÑ Á ! 5 5 initial condition.

F 5 and
5) satisfies the Lipschitz conditions, due to the boundedness of the coefficients , B 4 B and .Hence it has a unique strong solution.5 Lemma 3.3: Let resp.be the solution of resp.

4 . 3 : 4 . 4 :
continuous with respect to Lebesgue measure.Lemma The distributional derivative is the unique solution of the linear F µ > stochastic differential equation Ú Û Ü .Ð=ß >Ñ oe , Ð=ß \ Ñ Ð=ß >Ñ.=  Ð=ß \ Ñ Ð=ß >Ñ.measure, is well defined and does not depend on the versions of F µ Ð=ß >Ñ the Borel derivatives , .Moreover, since the coefficients and (4.2) satisfy Lipschitz conditions and has a unique -adapted continuous Y µ > solution.The fact that satisfies equation (4.2) is based on the absolute continuity of F µ Ð=ß >Ñ the law of and on the approximation of the coefficients and by smooth ones.See [2] The main result of this section is the following.Theorem Assume that the coefficients and the functional satisfy hypotheses ,ß P 5 Ð#Þ"Ñ Ð#Þ&Ñ À -.Then the following martingale representation formula holds PÐ\ Ñ oe IÒ Ð>ÑÎ Ó Ð>ß \ Ñ.F  I ÒPÐ\ ÑÓ T The proof of Theorem 4.4 is based on the next two lemmas.Lemma 4.5: Let be the smooth functions defined in the last section.Let the , dominated convergence theorem.Applying the standard arguments of stochastic differential equations yields

5 Lemma 4 . 6 :
Let  and   be the unique solutions of the following stochastic \ 5

>
According to Lemmas 4.5 and 4.6 and the continuity of , one can pass to the limit .Ð.=ß † Ñ in the last formula and obtain the desired result.Note that we have proved the convergence in Note that since the law of is absolutely continuous with respect to Lebesgue measure, \ B FÐ=ß >Ñ is well defined (the proof is performed as in Lemma 3.2).Now apply the Hausmann- .