Numerical Scheme for Backward Doublay Stochastic Differential Equations with Time Delayed Coefficients

In this paper, we present some assumptions to get the numerical scheme for backward doubly stochastic differential delay equations (shortly-BDSDDEs), and we propose a scheme of BDSDDEs and discuss the numerical convergence and rate of convergence of our scheme.


Introduction
Backward stochastic differential equations ( shortly-BSDEs) have been first presented in Pardoux and Peng [16,17] in order to proved existence and uniqueness of the adapted solutions and presented a new class of backward doubly stochastic differential equations, further investigations being (see [3,4,11,13]). A lot of mathematicians interested in a numerical methods for approximating solution of BSDEs ( see [1,10,14,15,18,22]). Xuerong Mao et al. [21] discussed the effects of environmental noise on the delay Lotka-Volterra model. Brahim Boufoussi et al. [2] presented a new class of backward doubly stochastic differential equations, this a new class depend on an integral with respect to an adapted continuous increasing process. Lukasz Delong [5,6] studied applications of a new class of time-delayed BSDEs and he gives examples of pricing, hedging and portfolio management problems which could be established in the framework of backward stochastic differential delay equation. Wen Lu et al. [19] investigated a class of multivalued backward doubly stochastic differential delay equation, and they proved the existence and uniqueness of the solutions for these equations under Lipschitz condition. Using the Euler-Maruyama method, Xiaotai Wu and Litan Yan [20] defined the numerical solutions of doubly perturbed stochastic delay differential equations driven by Levy process, and they proved the numerical solutions converge to the exact solutions with the local Lipschitz condition. Delong and Imkeller [7] presented a class of BSDEs with time delayed, and they established the existence and uniqueness of a solution for BSDEs with time delayed. Also, they [8] proved the existence and uniqueness as well as the Malliavin's differentiability of the solution for BSDEs with delayed time. Moreover, Diomande and Maticiuc [9] proved the existence and uniqueness of a solution for multivalued BSDEs with time delayed generators. Besides, Lu and Ren [12] established the existence and uniqueness of the solutions for a class of backward doubly stochastic differential equations with time delayed coefficients under Lipschitz condition. The purpose of this work is to study the numerical convergent of backward doubly stochastic delay differential equations ( shortly-BDSDDEs ) that has the following In our work, we extend the approach of BDSDDEs in the general case, and introduce some general assumptions on the numerical convergence of backward doubly stochastic differential equations with time delayed coefficients. Furthermore, we present a numerical scheme based on iterative regression functions which are approximated by projection on vector space of functions. Also, we discuss the numerical convergence and rate of convergence of BDSDDEs Lipschitz condition. The present paper is organized as follows: In section 2, we present some preliminaries that explain the approximation scheme for BDSDDEs. In section 3, we consider the approximation solution of BDSDDEs and prove some problems that useful for our work. In section 4, we have discussed the numerical convergence and rate of convergence of our scheme.

Notations, preliminaries and basic assumptions
In this section, we provide some assumptions and space used in the sequel. Therefore, we consider two independent standard d-dimensional Brownian motions , respectively. In this paper, we consider the following BDSDE with time delayed coefficients Assumption (H5): There exists a positive constant

A numerical scheme for BDSDDEs
In this section, we propose a numerical scheme is based upon a descretization of (1). Moreover, for all integers We can be approximated by the discrete equation . Therefore, we consider a class of BDSDDEs as the form

Main results
This section is devoted to discuss numerical convergence and rate of convergence of BDSDDEs.
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