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

Stochastic theta methods for random periodic solution of stochastic differential equations under non-globally Lipschitz conditions