Convergence of a spatial semi-discretization for a backward semilinear stochastic parabolic equation

TL;DR

This paper proves the convergence and spatial accuracy of a semi-discrete finite element method for backward semilinear stochastic parabolic equations, with applications to stochastic control.

Contribution

It establishes the first-order spatial convergence of a standard finite element method for these equations and applies the results to stochastic control problems.

Findings

Higher regularity of solutions is derived.

First-order spatial accuracy is achieved.

Application to stochastic linear quadratic control is demonstrated.

Abstract

This paper studies the convergence of a spatial semi-discretization for a backward semilinear stochastic parabolic equation. The filtration is general, and the spatial semi-discretization uses the standard continuous piecewise linear element method. Firstly, higher regularity of the solution to the continuous equation is derived. Secondly, the first-order spatial accuracy is derived for the spatial semi-discretization. Thirdly, an application of the theoretical result to a stochastic linear quadratic control problem is presented.