Efficient strong integrators for linear stochastic systems

TL;DR

This paper introduces advanced numerical schemes based on Magnus and Neumann expansions for solving linear stochastic differential equations, demonstrating improved accuracy and convergence properties, with applications in stochastic control.

Contribution

It develops and analyzes new strong integrators for linear stochastic systems, including convergence rates and error-cost relations, especially for non-commuting vector fields.

Findings

Magnus expansion-based methods outperform stochastic Taylor schemes in mean-square accuracy.

Explicit convergence rates are derived for multi-dimensional stochastic integrals.

Numerical studies demonstrate the effectiveness of the proposed methods in stochastic control applications.

Abstract

We present numerical schemes for the strong solution of linear stochastic differential equations driven by an arbitrary number of Wiener processes. These schemes are based on the Neumann (stochastic Taylor) and Magnus expansions. Firstly, we consider the case when the governing linear diffusion vector fields commute with each other, but not with the linear drift vector field. We prove that numerical methods based on the Magnus expansion are more accurate in the mean-square sense than corresponding stochastic Taylor integration schemes. Secondly, we derive the maximal rate of convergence for arbitrary multi-dimensional stochastic integrals approximated by their conditional expectations. Consequently, for general nonlinear stochastic differential equations with non-commuting vector fields, we deduce explicit formulae for the relation between error and computational costs for methods of arbitrary order. Thirdly, we consider the consequences in two numerical studies, one of which is an application arising in stochastic linear-quadratic optimal control.