The Numerical Invariant Measure of Stochastic Differential Equations With Markovian Switching.

The existence and uniqueness of the numerical invariant measure of the backward Euler-Maruyama method for stochastic differential equations with Markovian switching is yielded, and it is revealed that the numerical invariant measure converges to the underlying invariant measure in the Wasserstein metric. The global Lipschitz condition on the drift coefficients required by [J. Bao, J. Shao, and C. Yuan, Potential Anal., 44 (2016), pp. 707-727] and [X. Mao, C. Yuan, and G. Yin, J. Comput. Appl. Math., 174 (2005), pp. 1-27] is released. Under a polynomial growth condition imposed on drift coefficients we show that the convergence rate of the numerical invariant measure is polynomial. Several examples and numerical experiments are given to verify our theory.