The Numerical Invariant Measure of Stochastic Differential Equations With Markovian Switching

TL;DR

This paper establishes the existence, uniqueness, and convergence of the numerical invariant measure for stochastic differential equations with Markovian switching using the backward Euler-Maruyama method, relaxing previous Lipschitz conditions.

Contribution

It proves the convergence of the numerical invariant measure to the true measure and estimates the convergence rate under polynomial growth conditions, removing the need for global Lipschitz assumptions.

Findings

Numerical invariant measure converges to the true invariant measure in Wasserstein metric.

Convergence rate is estimated under polynomial growth conditions.

Numerical experiments verify the theoretical results.

Abstract

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. Under the polynomial growth condition of drift term the convergence rate is estimated. The global Lipschitz condition on the drift coefficients required by Bao et al., 2016 and Yuan et al., 2005 is released. Several examples and numerical experiments are given to verify our theory.