Invariant measure of numerical approximation to periodic stochastic differential equations with Markov switching

Numerical approximations to the invariant measures of a class of stochastic differential equations (SDEs) with periodic coefficients are studied in this work. Compared with those existing fruitful results on the numerical studies on the invariant measures of autonomous SDEs, to our best knowledge, this paper is the first one to devote to the case of nonautonomous SDEs that are extensively recognized to characterize various real-world problems in finance and biology. Technical challenges including time-inhomogeneity and periodicity make this paper a challenging and nontrivial work. The existence and uniqueness of the invariant measure of the numerical solution and its convergence to the underlying one in the Wasserstein distance are proved in this paper. Moreover, we obtain the exponential decay rate of the numerical solution to its invariant measure and the polynomial convergence rate of the numerical invariant measure to the underlying one. Numerical simulations are provided to illustrate those theoretical results.