A New Second Order Numerical Scheme for Solving Forward Backward Stochastic Differential Equations with Jumps

In this paper, we propose a new second order numerical scheme for solving backward stochastic differential equations with jumps with the generator ( ) ( ) ( ) t t t t f r t x y h t z g t , , = + + Γ linearly depending on t z . And we theoretically prove that the convergence rates of them are of second order for solving t y and of first order for solving t z and t Γ in p L norm.


( ) ( ) ( ) t t t t f r t x y h t z g t , ,
= + + Γ linearly de-

Introduction
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 [1], Peng obtained the relation between the backward stochastic differntial equation and the parabolic partial differential equation (PDE), and in Peng (1990), the stochastic maximum principle for optimal control problems were based on BSDEs.The applications of BSDEs now cover many scientific fields, such as stochastic control, stock markets, risk measure, turbulence fluid flow, biology, chemical reactions, partial differential equations, and so on.Thus it is very important and useful to obtain solutions of BSDEs for real applications.However, it is often quite difficult to obtain analytic solutions of BSDEs, so computing approximate solutions of BSDEs become highly desired, by using the relation between the BSDE and PDE.As far as we know, there have been very few schemes obtained with second-order convergence rate, such as [2] [3].
In this paper, we propose a new second order numerical scheme for the solution of forward-backward stochastic differential qquations (FBSDE in short) with jumps with the following form ( ) ( ) ( ) ( ) ( ) ( ) From [4], we know that the solution ( ) , , , x y z Γ can be represented as where the vector function ( ) , u t x is the classical solution of the following parabolic differential equation (PDE) of the form where x u ∇ denotes the gradient of u with respect to the space variable x,

Preliminaries and Notation
Let T be a fixed positive number and L -integrable, and satisfies (1).Under some standard conditions on the functions f and h, there is a unique adapted random process.Now we introduce a new probability space: for { } : 0 t t Λ ≥ is an exponential martingale and satisfies ( ) 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 ( ) , and the coefficient function

L f t x e f t x e a t x f t x e b t x b t x f t x e t x x x L f t x e b t x f t x e x L f t x e f t x c t x v e f t x e t T x R e ε
Now we introduce some basic notations.

• { } t t s T ≤ ≤
 : the σ-field generated by the Brownian motion.
• Throughout this paper, we denote by C a generic constant depending only on T, the upper bounds of the derivatives of the functions f.

Numerical Schemes for Solving BSDE
From the time interval [ ] 0,T , we introduce the following time partition: According to (1), it's easy to obtain that for 0 From ( 5) and (11),we have From (15), it is easy to obtain that for 0 Taking the conditional mathematical expectation on both side of the obtained equation, and by the nature of the conditional mathematical expectation,we deduce ( ) Based on (17), we have ( ) ( ) ( ) where and ( ) ( ) According to Lemma From ( 22) we have, where ( ) .
Scheme 1.Given ( ) 0 , , , x y z Γ , solve ( ) .However, all error estimate that we obtain in the sequel also hold for general multidimensional BSDEs.In our error analysis, we will use a constraint on the time partition step Let us introduce the following Lemma, its proof can be found in the reference [2].
) be the truncation errors defined in (21), ( 23) and (26), respectively.It holds that Here C is a positive constant depending on T. We first give the error estimate for Under the conditions of the theorem and by Lemma 2,we deduce that, ( ) ( ) ( ) where L is the Lipschitz constant of ( ) , , f t x y with respect to y. Applying the inequality ( ) ( ) ( ) + ∆ , and which by the inequality ( ) ( ) Taking the mathematical expectation on both sides of (35), for sufficiently small , the time step constraint (28) and the inequality ( ) , . , , . 1 , where C is a constant depending on T. is a positive number which depends on p and the constant C in Lemma 2. Taking the mathematical expectation on both sides of the Equation (42) gives