Euler time discretization of backward doubly SDEs and application to semilinear SPDEs

Abstract

This paper investigates a numerical probabilistic method for the solution of some semilinear stochastic partial differential equations (SPDEs in short). The numerical scheme is based on discrete time approximation for solutions of systems of decoupled forward-backward doubly stochastic differential equations. Under standard assumptions on the parameters, the convergence and the rate of convergence of the numerical scheme is proven. The proof is based on a generalization of the result on the path regularity of the backward equation.

An implicit numerical scheme for a class of backward doubly stochastic differential equations

“…We consider a numerical scheme similar to that introduced in [1]. For this scheme, we are able to achieve an estimate on the error in p-th moment, which is better than the estimates existing in the literature.…”

“…The presence of a backward Itô stochastic integral creates additional difficulties when deriving the path regularity of the process Z and computing the rate of convergence of numerical schemes. In the present paper, we consider an implicit numerical scheme introduced by Bachouch, Ben Lasmar, Matoussi and Mnif in a unpublished note [1]. Under the general assumptions on the terminal variable ξ and the generator f considered in [9] we have been able to show the Hölder continuity of the process Z and to derive a rate of convergence of the scheme (see the estimate (3.33)) in Theorem 3.9.…”