Order-one explicit approximations of random periodic solutions of semi-linear SDEs with multiplicative noise

TL;DR

This paper develops an explicit order-one approximation scheme for random periodic solutions of semi-linear SDEs with multiplicative noise, providing convergence analysis and numerical validation.

Contribution

It introduces a novel explicit scheme for approximating random periodic solutions of SDEs with multiplicative noise, with a new error analysis approach independent of high-order moment bounds.

Findings

Achieves expected order-one mean square convergence.

Validates theoretical results with numerical examples.

Provides a new method for analyzing error bounds without high-order moments.

Abstract

This paper is devoted to order-one explicit approximations of random periodic solutions to multiplicative noise driven stochastic differential equations (SDEs) with non-globally Lipschitz coefficients. The existence of the random periodic solution is demonstrated as the limit of the pull-back of the discretized SDE. A novel approach is introduced to analyze mean-square error bounds of the proposed scheme that does not depend on a prior high-order moment bounds of the numerical approximations. Under mild assumptions, the proposed scheme is proved to achieve an expected order-one mean square convergence in the infinite time horizon. Numerical examples are finally provided to verify the theoretical results.