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.

翻译：本文重新探讨了带加性噪声的半线性随机微分方程（SDEs）随机周期解的数值逼近中后向欧拉方法的应用。在相较于文献更宽松的条件下，讨论了随机周期解作为SDEs拉回流极限的存在唯一性。证明后向欧拉格式在均方意义下具有一阶收敛性，这改进了现有的一半阶收敛结果。最后通过数值算例验证了理论分析。