Invariant measures of the Milstein method for stochastic differential equations with commutative noise

Abstract

In this paper, the Milstein method is used to approximate invariant measures of stochastic differential equations with commutative noise. The decay rate of the transition probability kernel generated by the Milstein method to the unique invariant measure of the method is observed to be exponential with respect to the time variable. The convergence rate of the numerical invariant measure to the underlying counterpart is shown to be one. Numerical simulations are presented to demonstrate the theoretical results.