Random periodic solutions of SDEs: Existence, uniqueness and numerical issues

The aim of this paper is to discuss existence and uniqueness of random periodic solutions to stochastic differential equations (SDEs) with multiplicative noise under a one-sided Lipschitz condition, as well as on their numerical approximation via two classes of stochastic θ-methods, i.e., θ-Maruyama methods with θ∈[1/2,1] and θ-Milstein ones with θ∈[0,1]. The existence of the random periodic solutions as the limit of the pull-back flows of the discretized SDEs and the strong convergence rate of the aforementioned methods are also investigated. Selected numerical experiments confirming the theoretical analysis are also given.