Mean-square numerical approximations to random periodic solutions of stochastic differential equations

This paper is devoted to the possibility of mean-square numerical approximations to random periodic solutions of dissipative stochastic differential equations. The existence and expression of random periodic solutions are established. We also prove that the random periodic solutions are mean-square uniformly asymptotically stable, which ensures the numerical approximations are feasible. The convergence of the numerical approximations by the random Romberg algorithm is also proved to be mean-square. A numerical example is presented to show the effectiveness of the proposed method.