Numerical Approximation of Random Periodic Solutions of Stochastic Differential Equations

TL;DR

This paper develops and analyzes numerical schemes, specifically Euler-Maruyama and Milstein methods, for approximating random periodic solutions of stochastic differential equations with multiplicative noise, providing convergence rates and measure approximation results.

Contribution

It introduces a numerical approach to approximate random periodic solutions of SDEs and establishes convergence rates for these methods.

Findings

Euler-Maruyama scheme converges with rate √Δt in mean-square sense.

Milstein scheme converges with rate Δt in mean-square sense.

Weak convergence of the periodic measure is also established.

Abstract

In this paper, we discuss the numerical approximation of random periodic solutions (r.p.s.) of stochastic differential equations (SDEs) with multiplicative noise. We prove the existence of the random periodic solution as the limit of the pull-back flow when the starting time tends to $-\infty$ along the multiple integrals of the period. As the random periodic solution is not explicitly constructible, it is useful to study the numerical approximation. We discretise the SDE using the Euler-Maruyama scheme and moldiflied Milstein scheme. Subsequently we obtain the existence of the random periodic solution as the limit of the pull-back of the discretised SDE. We prove that the latter is an approximated random periodic solution with an error to the exact one at the rate of $\sqrt {\Delta t}$ in the mean-square sense in Euler-Maruyama method and $\Delta t$ in the Milstein method. We also obtain the weak convergence result for the approximation of the periodic measure.