The random periodic solution of a stochastic differential equation with a monotone drift and its numerical approximation

TL;DR

This paper investigates the existence, uniqueness, and numerical approximation of random periodic solutions for stochastic differential equations with monotone drifts, providing convergence rates for the backward Euler-Maruyama method.

Contribution

It establishes the existence and uniqueness of random periodic solutions under monotonicity conditions and analyzes the convergence of their numerical approximations.

Findings

Proves existence and uniqueness of random periodic solutions.

Establishes strong convergence rate of the backward Euler-Maruyama method.

Obtains weak convergence results for the periodic measure approximation.

Abstract

In this paper we study the existence and uniqueness of the random periodic solution for a stochastic differential equation with a one-sided Lipschitz condition (also known as monotonicity condition) and the convergence of its numerical approximation via the backward Euler-Maruyama method. The existence of the random periodic solution is shown as the limits of the pull-back flows of the SDE and discretized SDE respectively. We establish a convergence rate of the strong error for the backward Euler-Maruyama method and obtain the weak convergence result for the approximation of the periodic measure.