A projected Euler Method for Random Periodic Solutions of Semi-linear SDEs with non-globally Lipschitz coefficients

TL;DR

This paper introduces a projected Euler method for numerically approximating random periodic solutions of semi-linear SDEs with non-globally Lipschitz coefficients, demonstrating convergence and validating with numerical examples.

Contribution

The paper proposes a new explicit discretization scheme, the projected Euler method, for semi-linear SDEs with non-globally Lipschitz coefficients, establishing convergence without high-order moment bounds.

Findings

Mean square convergence rate of 0.5 for multiplicative noise

Convergence rate of 1 for additive noise

Numerical examples confirm theoretical results

Abstract

The present work introduces and investigates an explicit time discretization scheme, called the projected Euler method,to numerically approximate random periodic solutions of semi-linear SDEs under non-globally Lipschitz conditions. The existence of the random periodic solution is demonstrated as the limit of the pull-back of the discretized SDE. Without relying on a priori high-order moment bounds of the numerical approximations, the mean square convergence rate of the approximation scheme is proved to be order $0.5$ for SDEs with multiplicative noise and order $1$ for SDEs with additive noise. Numerical examples are also provided to validate our theoretical findings.