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A new class of reflected backward stochastic differential equations (RBSDEs) driven by Teugels martingales associated with Lévy process and Countable Brownian Motions are investigated. Via approximation, the existence and uniqueness of solution to this kind of RBSDEs are obtained.

Recently, Y. Ren [

where the

Note that, in all the previous works, the equations are driven by finite Brownian motions. In their recent work, Pengju Duan et al. [

where the dW is the standard forward stochastic Itô integral and the

Motivated by [

The paper is devoted to prove the existence and uniqueness of a solution for RBSDEs driven by a Lévy process and countable Brownian motions.

The paper is organized as follows. In section 2, we give some preliminaries and notations. In section 3, we establish the main results.

Throughout this paper, T is a positive constant and

measure

Let

where for any process

Note that

collection

Let us introduce some spaces:

・

・

・

・

・

We will denote by

In the sequel, for ease of notation, we set

Furthermore, we denote by

where

In [

Definition 2.1. A solution of a (1.2) is a triplet of

1)

2)

3) K is a continuous and increasing process with

Throughout the paper, we let the coefficients

(H1) for all

(H2) for all

where

(H3)

(H4) S is a real-valued, càdàg process such that

with

stopping times (see He et al. [

We first establish the existence and uniqueness result for RBSDEs driven by finite Brownian motions and a Lévy process:

For any

Lemma 3.2. Assume (H1) - (H4). Then, there exists a unique solution

Proof. For

respectively by

Now, we are ready to establish the main result of this paper which is the following theorem.

Theorem 3.3. Under assumptions (H1)-(H4), there exists a unique solution

Proof. (Existence.) By Lemma 3.1, for any

The idea consists to study the convergence of the sequence

where

From assumption (H2) and Young’s inequality, for any

Using again Young inequality, we have for any

Since

we have, for any

Therefore,

Consequently,

We choose

Applying Gronwall’s inequality, we get

Therefore, we have the existence of a constant

which by Burkhölder-Davis-Gundy’s inequality provides

Now, we show that

Applying Itô’s formula to

Taking expectation in both side of (3.5) and noting that

Using again Young’s inequality, assumption (H2) and the estimates (3.3), we obtain,

where

Therefore, by Gronwall’s inequality, we have

which, by Burkholder-Davis-Gundy inequality provides

Well, from assumptions (H1)-(H2), we have

Consequently, we get,

Moreover, from (3.4) together with Hölder’s and Burkholder-Davis-Gundy’s inequalities, we have

which, together with assumption (H2) and (3.7), provides

Consequently,

Now, let us show that the process

Also, by Burkhölder-Davis-Gundy’s inequality, we get

and

Now, from (H1)-(H2) and the fact that

which implies that

Moreover,

Therefore,

On the other hand, from the result of Saisho [

Finally, passing to the limit in (3.2), we conclude that

(Uniqueness.) Let

Applying Itô’s formula to

Taking expectation in both side of (3.10) and noting that

Using again Young’s inequality

Choosing

On the other hand, since,

we have

Jean-MarcOwo, (2015) Reflected BSDEs Driven by Lévy Processes and Countable Brownian Motions. Applied Mathematics,06,2240-2247. doi: 10.4236/am.2015.614197