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 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.

翻译：本文提出并研究了一种显式时间离散化格式——投影欧拉方法，用于在非全局Lipschitz条件下数值逼近半线性随机微分方程的随机周期解。通过离散化SDE的回拉极限证明了随机周期解的存在性。在不依赖数值解先验高阶矩界的条件下，证明了乘性噪声SDE的均方收敛阶为0.5，加性噪声SDE的均方收敛阶为1。数值算例进一步验证了理论结果。