Temporal semi-discretizations of a backward semilinear stochastic evolution equation

TL;DR

This paper investigates the convergence properties of three different temporal semi-discretization methods for backward semilinear stochastic evolution equations, providing theoretical guarantees and an application to stochastic control problems.

Contribution

It establishes convergence results for three semi-discretization schemes, including explicit rates for the third scheme and its application to stochastic linear quadratic control.

Findings

Convergence of the first two semi-discretizations under Lipschitz conditions.

Explicit convergence rate derived for the third semi-discretization.

Application of the third scheme to approximate optimal control in stochastic systems.

Abstract

This paper studies the convergence of three temporal semi-discretizations for a backward semilinear stochastic evolution equation. For general terminal value and general coefficient with Lipschitz continuity, the convergence of the first two temporal semi-discretizations is established, and an explicit convergence rate is derived for the third temporal semi-discretization. The third temporal semi-discretization is applied to a general stochastic linear quadratic control problem, and the convergence of a temporally semi-discrete approximation to the optimal control is established.