Malliavin differentiability of solutions of SPDEs with L\'evy white noise

In this article, we consider a stochastic partial differential equation (SPDE) driven by a L\'evy white noise, with Lipschitz multiplicative term $\sigma$. We prove that under some conditions, this equation has a unique random field solution. These conditions are verified by the stochastic heat and wave equations. We introduce the basic elements of Malliavin calculus with respect to the compensated Poisson random measure associated with the L\'evy white noise. If $\sigma$ is affine, we prove that the solution is Malliavin differentiable and its Malliavin derivative satisfies a stochastic integral equation.


Introduction
In this article, we consider the stochastic partial differential equation (SPDE): L ( , ) = ( ( , ))̇( , ) , ∈ [0, ] , ∈ R, (1) with some deterministic initial conditions, where L is a second-order differential operator on [0, ]×R,̇denotes the formal derivative of the Lévy white noise (defined below), and the function : R → R is Lipschitz continuous. A process = { ( , ); ∈ [0, ], ∈ R} is called a (mild) solution of (1) if is predictable and satisfies the following integral equation: where is the solution of the deterministic equation L = 0 with the same initial conditions as (1) and is the Green function of the operator L.
The study of SPDEs with Gaussian noise is a welldeveloped area of stochastic analysis, and the behaviour of random field solutions of such equations is well understood. We refer the reader to [1] for the original lecture notes which led to the development of this area and to [2,3] for some recent advances. In particular, the probability laws of these solutions can be analyzed using techniques from Malliavin calculus, as described in [4,5].
On the other hand, there is a large literature dedicated to the study of stochastic differential equations (SDE) with Lévy noise, the monograph [6] containing a comprehensive account on this topic. One can develop also a Malliavin calculus for Lévy processes with finite variance, using an analogue of the Wiener chaos representation with respect to underlying Poisson random measure of the Lévy process. This method was developed in [7] with the same purpose of analyzing the probability law of the solution of an SDE driven by a finite variance Lévy noise. More recently, Malliavin calculus for Lévy processes with finite variance has been used in financial mathematics, the monograph [8] being a very readable introduction to this topic.
There are two approaches to SPDEs in the literature. One is the random field approach which originates in John Walsh's lecture notes [1]. When using this approach, the solution is viewed as a real-valued process which is indexed by time and space. The other approach is the infinite-dimensional approach, due to Da Prato and Zabczyk [9], according to 2 International Journal of Stochastic Analysis which the solution is a process indexed by time only, which takes values in an infinite-dimensional Hilbert space. It is not always possible to compare the solutions obtained using the two approaches (see [10] for several results in this direction). SPDEs with Lévy noise were studied in the monograph [11], using the infinite-dimensional approach. In the present article, we use the random field approach for examining an SPDE driven by the finite variance Lévy noise introduced in [12], with the goal of studying the Malliavin differentiability of the solution. As mentioned above, this study can be useful for analyzing the probability law of the solution. We postpone this problem for future work.
We begin by recalling from [12] the construction of the Lévy white noise driving (1). We consider a Poisson random measure (PRM) on the space In addition, we assume that ] satisfies the following condition: We denote bŷthe compensated PRM defined bŷ ( ) = ( ) − ( ) for any ∈ U with ( ) < ∞, where U is the class of Borel sets in . We denote by F the -field generated by ([0, ] × × Γ) for all ∈ [0, ], ∈ B (R), and Γ ∈ B (R 0 ). We denote by B (R) the class of bounded Borel sets in R and by B (R 0 ) the class of Borel sets in R 0 which are bounded away from 0.
(iii) For any 0 < ≤ and for any ∈ B (R), ( ) − ( ) is independent of F and has characteristic function We denote by F the -field generated by ( ) for all ∈ [0, ]. For any ℎ ∈ 2 ([0, ] × R), we define the stochastic integral of ℎ with respect to : Using the same method as in Itô's classical theory, this integral can be extended to random integrands, that is, to the class of predictable processes = { ( , ); ∈ [0, ], ∈ R}, such that ∫ 0 ∫ R | ( , )| 2 < ∞. The integral has the following isometry property: Recall that a process = { ( , ); ≥ 0, ∈ R } is predictable if it is measurable with respect to the predictable -field on Ω × R + × R, that is, the -field generated by elementary processes of the form where 0 < < , is a bounded and F -measurable random variable, and ∈ B (R). This article is organized as follows. In Section 2, we introduce the basic elements of Malliavin calculus with respect to the compensated Poisson random measurê. In Section 3, we prove that, under a certain hypothesis, (1) has a unique solution. This hypothesis is verified in the case of the wave and heat equations. In Section 4, we examine the Malliavin differentiability of the solution, in the case when the function is affine. Finally, in the Appendix, we include a version of Gronwall's lemma which is needed in the sequel.

Malliavin Calculus on the Poisson Space
In this section, we introduce the basic ingredients of Malliavin calculus with respect tô, following very closely the approach presented in Chapters 10-12 of [8]. The difference compared to [8] is that our parameter space has variables ( , , ) instead of ( , ). For the sake of brevity, we do not include the proofs of the results presented in this section. These proofs can be found in Chapter 6 of the doctoral thesis [13] of the second author.
We set H = 2 ( , U, ) and H ⊗ = 2 ( , U , ). We denote by H ⊙ the set of all symmetric functions ∈ H ⊗ . We denote by H C , H ⊗ C , H ⊙ C the analogous spaces of C-valued functions.
International Journal of Stochastic Analysis we define the -fold iterated integral of with respect tôby where ∈ H ⊙ for all ≥ 1 and 0 = ( ).
The chaos expansion plays a crucial role in developing the Malliavin calculus with respect tô. In particular, the Skorohod integrals with respect tôand are defined as follows.
For each ∈ , let ( ) = ∑ ≥0 ( (⋅, )) be the chaos expansion of ( ), with (⋅, ) ∈ H ⊙ . One denotes bỹ( 1 , . . . , , ) the symmetrization of with respect to all + 1 variables. One says that is Skorohod integrable with respect tô(and one writes ∈ Dom( )) if In this case, one defines the Skorohod integral of with respect tôby ∈ R} be a squareintegrable process such that ( , ) is F -measurable for any ∈ [0, ] and ∈ R. One says that is Skorohod integrable with respect to (and one writes ∈ Dom( )) if the process { ( , ) ; ( , , ) ∈ } is Skorohod integrable with respect tô. In this case, one defines the Skorohod integral of with respect to by The following result shows that the Skorohod integral can be viewed as an extension of the Itô integral.
We now introduce the definition of the Malliavin derivative.
Definition 3. Let ∈ 2 (Ω) be an F -measurable random variable with the chaos expansion = ∑ ≥0 ( ) with ∈ H ⊙ . One says that is Malliavin differentiable with respect tôif In this case, one defines the Malliavin derivative of with respect tôby One denotes by D 1,2 the space of Malliavin differentiable random variables with respect tô.
The following result shows that the Malliavin derivative is a difference operator with respect tô, not a differential operator.
Similar to the Gaussian case, we have the following results.

Existence of Solution
In this section, we show that (1) has a unique solution. We recall that is the solution of the homogeneous equation L = 0 with the same initial conditions as (1), and is the Green function of the operator L on R + × R. We assume that, for any ∈ [0, ], ( , ⋅) ∈ 1 (R) and we denote by F ( , ⋅) its Fourier transform: We suppose that the following hypotheses hold.
Existence. We use the same argument as in the proof of Theorem 13 of [16]. We denote by ( ) ≥0 the sequence of Picard iterations defined by 0 ( , ) = ( , ) and +1 ( , ) = ( , ) By induction on , it can be proved that the following properties hold: Hypotheses (H1) and (H2) are needed for the proof of property (iii). From properties (iii) and (iv), it follows that has a predictable modification, denoted also by . This modification is used in definition (31) of +1 ( , ). Using the isometry property (8) of the stochastic integral and (28), we have where Taking the supremum over ∈ R in the previous inequality, we obtain that for any ∈ [0, ] and ≥ 0. By applying Lemma A.1 (the Appendix) with = 0 and = 2, we infer that This shows that the sequence ( ) ≥0 converges in 2 (Ω) to a random variable ( , ), uniformly in [0, ] × R; that is, To see that is a solution of (1), we take the limit in 2 (Ω) as → ∞ in (31). In particular, this argument shows that Uniqueness. Let ( ) = sup ∈R | ( , ) − ( , )| 2 , where and are two solutions of (1). A similar argument as above shows that

Malliavin Differentiability of the Solution
In this section, we show that the solution of (1) is Malliavin differentiable and its Malliavin derivative satisfies a certain integral equation. For this, we assume that the function is affine.
Our first result shows that the sequence of Picard iterations is Malliavin differentiable with respect tôand the corresponding sequence of Malliavin derivatives is uniformly bounded in 2 (Ω; H). Proof.
Step 1. We prove that the following property holds for any ≥ 0: For this, we use an induction argument on . Property (Q) is clear for = 0. We assume that it holds for and we prove that it holds for + 1.
We are now ready to state the main result of the present article.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.