SPDIEs and BSDEs Driven by Lévy Processes and Countable Brownian Motions

The paper is devoted to solving a new class of backward stochastic differential equations driven by Lévy process and countable Brownian motions. We prove the existence and uniqueness of the solutions to the backward stochastic differential equations by constructing Cauchy sequence and fixed point theorem. Moreover, we give a probabilistic solution of stochastic partial differential integral equations by means of the solution of backward stochastic differential equations. Finally, we give an example to illustrate.


Introduction
The backward stochastic differential equations (BSDEs for short), in the nonlinear cases, were firstly introduced by Pardoux and Peng [1] in order to give a probabilistic interpretation for the solution of semilinear parabolic partial differential equations.In the past decades, the equations have been extensively considered because of the applications in mathematic finance [2,3], stochastic games [4][5][6], and partial differential equations (PDEs for short) [7][8][9][10].
As the applications developed, different settings of BSDEs have been introduced.Pardoux and Peng [11] proposed a new class of BSDEs driven by two Brownian motions, which are called backward doubly stochastic differential equations (BDSDEs for short), in order to give a probabilistic interpretation for the solution of quasi-linear stochastic partial differential equations (SPDEs for short).Since then, many authors discussed various settings of BDSDEs, for example, Bally and Matoussi [12], Matoussi and Scheutzow [13], Zhang and Zhao [14,15], and the references therein.
In 2000, Nualart and Schoutens [16] gave a martingale representation of Lévy process.Furthermore, they [17] discussed the BSDEs driven by Lévy process and the application in finance.Following it, many authors were devoted to the BSDEs driven by Lévy process.Bahlali et al. [18] generalized the results [17] to the BSDEs driven by Teugels martingales associated with Lévy process and a Brownian motion.Also, they gave the application in partial differential integral equations (PDIEs for short).Ren et al. [19] introduced a class of BDSDEs driven by Teugels martingales associated with Lévy process and two Brownian motions.They obtained the existence and uniqueness of solution and gave the probabilistic interpretation for solutions of stochastic partial differential integral equations (SPDIEs for short).Later, Hu and Ren [20] discussed BDSDEs driven by Teugels martingales associated with Lévy process and an adapted continuous increasing process.Recently, Duan et al. [21] made further discussion of reflected backward stochastic differential equations driven by countable Brownian motions under Lipschitz conditions.Owo [22] studied the equations with continuous coefficients.
To the best of our knowledge, there are no works on the BSDEs driven by Teugels martingales associated with Lévy processes and countably many Brownian motions.Thus, we will make the first attempt to study such problem in this paper.
The structure of this paper is organized as follows.In Section 2, we present some basic notions and assumptions.Section 3 is devoted to the existence and uniqueness of solutions for BSDEs driven by Teugels martingales associated with Lévy processes and countably many Brownian motions by means of martingale representation theorem, fixed point theorem, and constructing Cauchy sequence.In Section 4, we discuss the connection between the BSDEs and SPDIEs.
Let N denote the totality of -null sets of F. For each  ∈ [0, ], we define where for any process   , Let us introduce some spaces which will be carried out in the following parts.
(ii) H 2 denotes the space of R-valued, square integrable, and F  -progressively measurable processes {  :  ∈ [0, ]} such that And we denote by P 2 the subspace of H 2 formed by the predictable processes.
(iii) S 2 denotes the set of R-valued, F  -measurable processes {  :  ∈ [0, ]} such that Let  2 be the space of R-valued sequences ) and P 2 ( 2 ) denote the corresponding space of  2 -valued processes endowed with the norm Now, we give the definition of the Teugels martingales denoted by { ()   }, associated with the Lévy processes {  ,  ∈ [0, ]}, which is given by where 1 ] for all  ≥ 1 and  ()   are power-jump processes.That is,
In this paper, we will discuss the following backward stochastic differential equations driven by Lévy process and countably many Brownian motions: where the integral with respect to   () is the classical backward Itô integral and the integral with respect to  ()  is standard forward Itô integral.
With the above preparation, we introduce the definition of solution of (8).
In order to get the solution of ( 8), we propose the following assumptions: (H3) There exist some nonnegative constants ,   ,   with Our conclusions depend on the extensive Itô formula in [19].

Existence and Uniqueness
In this section, we begin with establishing the existence and uniqueness of (8) in the case that  and  do not depend on  and  with finite noise; that is, Theorem 3. Assume that (H1)-(H3) hold.Then, there exists a unique solution (  ,   ) ∈ S 2 × P 2 ( 2 ) satisfying (14).
we set the filtration {C  :  ∈ [0, ]} as follows: and the C  -square integrable martingale is as follows: By the predictable representation property, there exists  ∈ P 2 ( 2 ) such that So, we have Let Hence, we have From the above equality, we can deduce the existence of solution of (14).The proof of uniqueness is a procedure similar to that in [11]; we omit it.
Proof.From Theorem 3, for each (  ,   ), there exists (  ,   ) satisfying Following it, we define a map Φ from S 2 ×P 2 ( 2 ) to itself; that is, Φ(  ,   ) = (  ,   ).In the following parts, we will show that Φ is a strict contraction with the norm for suitable constant  > 0. In addition, S Taking mathematical expectation on both sides, we obtain In the following part, we claim that (   ,    ) is Cauchy sequence in S 2 × P 2 ( 2 ).Applying Itô formula to |   −    | 2 , without loss of generality, we let  < ; then Taking mathematical expectation on both sides, we have By the Gronwall inequality and B-D-G inequality, we have Denote its limit by (  ,   ); from the continuity of  and  and Lebesgue dominated convergence theorem, we can imply that it is the solution of (8).
Uniqueness.We set If we define Ψ   ()/ = 2, when  = 0, then, 0 ≤ Ψ Taking expectation on both sides, Let  → ∞,  is large enough, and we have So, we complete the proof of uniqueness.

Application to SPDIEs
In this section, we consider the application of BSDEs driven by Lévy processes and countably many Brownian motions to the solution of a class of SPDIEs.Suppose that our Lévy processes have the form of   =  + ∫ ||<1 ((  (⋅, )) − ]()), where   (, ) denotes the random measure such that ∫ Λ   (⋅, ) is a Poisson process with parameter ](Λ) for all the set Λ where 0 ∉ Λ.Consider the following SDE: Under adequate conditions, there exists a unique solution of (40).
In order to get the main result, we give a technical lemma that appears in [17].