Order-One Convergence of the Backward Euler Method for Random Periodic Solutions of Semilinear SDEs

In this paper, we revisit the backward Euler method for numerical approximations of random periodic solutions of semilinear SDEs with additive noise. The existence and uniqueness of the random periodic solution are discussed as the limit of the pull-back flows of SDEs under a relaxed condition compared to literature. The backward Euler scheme is proved to converge with an order one in the mean square sense, which improves the existing order-half convergence. Numerical examples are presented to verify our theoretical analysis.