Numerical Computation for Backward Doubly SDEs with random terminal time

TL;DR

This paper develops numerical methods for backward doubly stochastic differential equations with random terminal times, aiming to solve related semilinear SPDEs and analyze approximation errors.

Contribution

It introduces a numerical scheme for BDSDEs with random terminal times, especially for first exit times, including error bounds and Euler approximation analysis.

Findings

Euler scheme for BDSDEs with random terminal time established

Error bounds for discrete approximation provided

Application to Dirichlet problems for semilinear SPDEs demonstrated

Abstract

In this article, we are interested in solving numerically backward doubly stochastic differential equations (BDSDEs) with random terminal time tau. The main motivations are giving a probabilistic representation of the Sobolev's solution of Dirichlet problem for semilinear SPDEs and providing the numerical scheme for such SPDEs. Thus, we study the strong approximation of this class of BDSDEs when tau is the first exit time of a forward SDE from a cylindrical domain. Euler schemes and bounds for the discrete-time approximation error are provided.