  In this paper we study the existence and uniqueness of the random periodic
solution for a stochastic differential equation with a one-sided Lipschitz
condition (also known as monotonicity condition) and the convergence of its
numerical approximation via the backward Euler-Maruyama method. The existence
of the random periodic solution is shown as the limits of the pull-back flows
of the SDE and discretized SDE respectively. We establish a convergence rate of
the strong error for the backward Euler-Maruyama method and obtain the weak
convergence result for the approximation of the periodic measure.
