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：本研究聚焦于随机微分方程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值在(1/2,1]范围内的随机theta方法生成的数值逼近解存在唯一随机周期解。研究表明，对于任意步长，每种ST方法的随机周期解均以均方意义强收敛于原SDE的随机周期解。具体而言，对于乘性噪声SDE的均方收敛阶为1/2，对于加性噪声SDE的均方收敛阶为1。最后通过数值实验验证了理论结果。