Euler time discretization of Backward Doubly SDEs and Application to Semilinear SPDEs

TL;DR

This paper develops a numerical scheme based on Euler time discretization for solving semilinear SPDEs via backward doubly stochastic differential equations, proving convergence and convergence rate under standard assumptions.

Contribution

It introduces a novel Euler discretization approach for backward doubly SDEs and applies it to semilinear SPDEs, with proven convergence and rate results.

Findings

Convergence of the numerical scheme is established.

The rate of convergence is quantified.

The method is applicable under standard assumptions.

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.