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In this paper, we revisit the backward Euler method for numericalapproximations of random periodic solutions of semilinear SDEs with additivenoise. Improved $L^{p}$-estimates of the random periodic solutions of theconsidered SDEs are obtained under a more relaxed condition compared toliterature. The backward Euler scheme is proved to converge with an order onein the mean square sense, which also improves the existing order-halfconvergence. Numerical examples are presented to verify our theoreticalanalysis.

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