Approximation of Invariant Measures for Regime-Switching Diffusions

TL;DR

This paper investigates the long-term behavior of Euler-Maruyama schemes for regime-switching diffusions, establishing conditions for the existence, uniqueness, and convergence of numerical invariant measures across different state space types.

Contribution

It introduces new methods for proving the existence and uniqueness of numerical invariant measures for regime-switching diffusions, including finite and countable state spaces, using Perron-Frobenius, eigenvalue, and M-Matrix theories.

Findings

Numerical invariant measures exist and are unique under specified conditions.

These measures converge in Wasserstein metric to the true invariant measures.

Examples demonstrate the practical applicability of the theoretical results.

Abstract

In this paper, we are concerned with long-time behavior of Euler-Maruyama schemes associated with a range of regime-switching diffusion processes. The key contributions of this paper lie in that existence and uniqueness of numerical invariant measures are addressed (i) for regime-switching diffusion processes with finite state spaces by the Perron-Frobenius theorem if the "averaging condition" holds, and, for the case of reversible Markov chain, via the principal eigenvalue approach provided that the principal eigenvalue is positive; (ii) for regime-switching diffusion processes with countable state spaces by means of a finite partition method and an M-Matrix theory. We also reveal that numerical invariant measures converge in the Wasserstein metric to the underlying ones. Several examples are constructed to demonstrate our theory.