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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.
The backward stochastic differential equations (BSDEs for short), in the nonlinear cases, were firstly introduced by Pardoux and Peng  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 [4–6], and partial differential equations (PDEs for short) [7–10].
As the applications developed, different settings of BSDEs have been introduced. Pardoux and Peng  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 , Matoussi and Scheutzow , Zhang and Zhao [14, 15], and the references therein.
In 2000, Nualart and Schoutens  gave a martingale representation of Lévy process. Furthermore, they  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.  generalized the results  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.  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  discussed BDSDEs driven by Teugels martingales associated with Lévy process and an adapted continuous increasing process. Recently, Duan et al.  made further discussion of reflected backward stochastic differential equations driven by countable Brownian motions under Lipschitz conditions. Owo  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 be a fixed terminal time. Let be a complete probability space, let and be mutually independent processes, where is a sequence of -valued standard Brownian motion and mutually independent, and is -valued Lévy process corresponding to a standard Lévy measure such that .
Let denote the totality of -null sets of . For each , we define where for any process .
Let us introduce some spaces which will be carried out in the following parts. (i) denotes the set of all -measurable random variables such that . (ii) denotes the space of -valued, square integrable, and - progressively measurable processes such that And we denote by the subspace of formed by the predictable processes.(iii) denotes the set of -valued, -measurable processes such that
Let be the space of -valued sequences such that . and denote the corresponding space of -valued processes endowed with the norm
Now, we give the definition of the Teugels martingales denoted by , associated with the Lévy processes , which is given by where for all and are power-jump processes. That is, and for , where and . The coefficients correspond to the orthonormalization of the polynomials with respect to the measure : We set
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: (H1). (H2)The functions and are progressively measurable such that (H3)There exist some nonnegative constants with and such that, for any ,
Our conclusions depend on the extensive Itô formula in .
Lemma 2. Let and such that Then Noting that , we have
3. 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 satisfying (14).
Proof. For , , we set the filtration as follows: and the -square integrable martingale is as follows: By the predictable representation property, there exists 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 ; we omit it.
With the preparation of above, we consider the following BSDEs with finite noise:
Theorem 4. Assume that (H1)–(H3) hold. Then, there exists a unique solution satisfying (21).
Proof. From Theorem 3, for each , there exists satisfying Following it, we define a map from to itself; that is, . In the following parts, we will show that is a strict contraction with the norm for suitable constant . In addition, is a Banach space.
Set , , where and are the solutions of (22) associated with and , respectively. Let . Applying Itô formula to , we have Taking mathematical expectation on both sides, we obtain With the conditions of (H1)–(H3), it follows that Furthermore, we have Let , , and , and we have Moreover, That is, It follows that is a strict contraction with the norm . Then, from B-D-G inequality, has a unique fixed point , which is the unique solution of (21).
Theorem 5. Under the conditions (H1)–(H3), there exists a unique solution satisfying (8).
Proof (existence). From Theorem 4, for each , there exists a unique solution of (21) under the conditions (H1)–(H3) denoted by : In the following part, we claim that is Cauchy sequence in . Applying Itô formula to , without loss of generality, we let ; then Taking mathematical expectation on both sides, we have Therefore, we obtain 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 , when , then, . Let be two solutions of (8); we apply Itô formula to , where , and is constant: Taking expectation on both sides, Let , is large enough, and we have So, we complete the proof of uniqueness.
4. 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 , where denotes the random measure such that is a Poisson process with parameter for all the set where .
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 .
Lemma 6. Let be a measurable function such that where is a nonnegative predictable process such that . Then, for each , we have
Consider the following BSDEs driven by Lévy processes and countably many Brownian motions: where .
Define where is the solution of the following SPDIEs: with , , and and for
In order to give the meaning of , we write the above SPDIEs in the following integral form: Suppose that is function that and are bounded by a polynomial function of , uniformly in . Next, we give the main result of this section.
Theorem 7. The unique adapted solution of (43) is given by
In the following, we give an example of SPDIEs.
Example 8. Suppose that the Lévy process has the form of , where is a sequence of independent Poisson processes with parameters . Its Lévy measure is , where denotes the positive mass measure at of size 1. Moreover, we assume that . Note that and , (see ). Let be the solution of the following equation: Then where is the solution of the following SPDIEs:
The author declares that he has no competing interests.
The work is partially supported by the NSF of China (10901003), key program of study home and abroad for young scholar sponsored by Anhui Province (gxfxZD2016261), the project for outstanding academic and technical backbone of Suzhou University (2014xjgg05), and Natural Science Foundation of Anhui Province (1508085MA10).
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