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In this paper, we propose a new second order numerical scheme for solving backward stochastic differential equations with jumps with the generator linearly depending on . And we theoretically prove that the convergence rates of them are of second order for solving and of first order for solving and in norm.

Bismut (1973) studied the existence of the linear backward stochastic differential equation, the results could be regarded as a promotion of a famous Girsanov theorem. The existence and uniqueness of solutions for nonlinear backward stochastic differential equations (BSDEs) were first proved by Pardoux and Peng (1990).Since then, BSDEs have been extensively studied by many researchers. In [

In this paper, we propose a new second order numerical scheme for the solution of forward-backward sto- chastic differential qquations (FBSDE in short) with jumps with the following form

From [

where the vector function

where

Let T be a fixed positive number and

defined a standard Brownian motion

A process

Now we introduce a new probability space: for

Let us first introduce the following lemma.

Lemma 1. Given the time partition

We use the following Itô-Taylor approximation to solve the forward SDEs with jumps

where

and the coefficient function

with

Now we introduce some basic notations.

・

・ Throughout this paper, we denote by C a generic constant depending only on T, the upper bounds of the derivatives of the functions f.

From the time interval

From (5) and (11),we have

From (12), (13) and (14), by applying Itô formula to

From (15), it is easy to obtain that for

Taking the conditional mathematical expectation

Based on (17), we have

where

and

According to Lemma 1, the equality

Let

From (22) we have,

where

Let

From (25) we have,

where

Based on (21), (23) and (26), for solving the BSDE (1) we propose the following scheme.

Scheme 1. Given

In this section, we will give the error estimates of Scheme 1 proposed in Section 3. Now we introduce the error

solution of the FBSDEs (1), and

Let us introduce the following Lemma, its proof can be found in the reference [

Lemma 2. Let

Here C is a positive constant depending on T. We first give the error estimate for

Theorem 1. Let

for

Proof. Let

Under the conditions of the theorem and by Lemma 2,we deduce that,

where L is the Lipschitz constant of

which by the inequality

Taking the mathematical expectation on both sides of (35), for sufficiently small

for

lead to

Then we turn to estimating the error

Theorem 2. Let

for

Proof. Let

By Lemma 2, the inequality

where

by using Theorem 1 and constraint (28), leads to

At last, we estimate the error

Theorem 3. Let

for

Proof. Let

By Lemma 2, the inequality

where

by using Theorem 1 and constraint (28), leads to

Hongqiang Zhou,Yang Li,Zhe Wang, (2016) A New Second Order Numerical Scheme for Solving Forward Backward Stochastic Differential Equations with Jumps. Applied Mathematics,07,1408-1414. doi: 10.4236/am.2016.712121