A Semidiscrete Galerkin Scheme for Backward Stochastic Parabolic Differential Equations

TL;DR

This paper introduces a numerical scheme combining Galerkin, truncation, and backward Euler methods to solve backward stochastic parabolic PDEs, providing a global $L^2$ error estimate.

Contribution

It develops a novel semidiscrete Galerkin scheme for backward stochastic parabolic PDEs with proven error bounds.

Findings

Global $L^2$ error estimate established

Effective approximation of backward stochastic PDEs achieved

Method demonstrated for initial-boundary value problems

Abstract

In this paper, we present a numerical scheme to solve the initial-boundary value problem for backward stochastic partial differential equations of parabolic type. Based on the Galerkin method, we approximate the original equation by a family of backward stochastic differential equations (BSDEs, for short), and then solve these BSDEs by the time discretization. Combining the truncation with respect to the spatial variable and the backward Euler method on time variable, we obtain the global $L^2$ error estimate.