Convergence of Finite Element Methods for Singular Stochastic Control

TL;DR

This paper introduces a finite element-based numerical method for solving singular stochastic control problems, providing convergence analysis and demonstrating effectiveness through examples with long-term average costs.

Contribution

It develops a finite element approximation approach for singular stochastic control problems and offers a detailed convergence analysis.

Findings

The method converges under specified assumptions.

Numerical examples validate the approach.

Effective for long-term average cost problems.

Abstract

A numerical method is proposed for a class of stochastic control problems including singular behavior. This method solves an infinite-dimensional linear program equivalent to the stochastic control problem using a finite element type approximation, which results in a solvable finite-dimensional program. The discretization scheme as well as the necessary assumptions are discussed, and a detailed convergence analysis for the discretization scheme is given. Its performance is illustrated by two examples featuring a long-term average cost criterion.