This work focuses on the numerical approximations of random periodic solutions of stochastic differential equations (SDEs). Under non-globally Lipschitz conditions, we prove the existence and uniqueness of random periodic solutions for the considered equations and its numerical approximations generated by the stochastic theta (ST) methods with theta within (1/2,1]. It is shown that the random periodic solution of each ST method converges strongly in the mean square sense to that of SDEs for all step size. More precisely, the mean square convergence order is 1/2 for SDEs with multiplicative noise and 1 for SDEs with additive noise. Numerical results are finally reported to confirm these theoretical findings.

翻译：本文关注随机微分方程(SDEs)随机周期解的数值逼近问题。在非全局Lipschitz条件下，我们证明了所考虑方程及其由theta属于(1/2,1]的随机theta(ST)方法生成的数值逼近的随机周期解的存在唯一性。研究表明，对于所有步长，每个ST方法的随机周期解在均方意义下强收敛于SDEs的随机周期解。更精确地，对于乘性噪声的SDEs，均方收敛阶为1/2，对于加性噪声的SDEs，均方收敛阶为1。最后，数值结果验证了这些理论发现。