Almost Sure Asymptotic Stability for Regime-Switching Diffusions

TL;DR

This paper investigates the long-term stability of regime-switching diffusions, providing new criteria for almost sure asymptotic stability across finite and countable state spaces, with applications to linear models.

Contribution

It introduces novel methods for establishing stability of regime-switching diffusions using Perron-Frobenius, eigenvalue, and M-Matrix theories, covering finite and countable state spaces.

Findings

Established stability criteria for finite state space diffusions using Perron-Frobenius theorem.

Derived stability conditions for reversible Markov chain-based diffusions via principal eigenvalues.

Extended stability analysis to countable state space diffusions using partition and M-Matrix techniques.

Abstract

In this paper, we discuss long-time behavior of sample paths for a wide range of regime-switching diffusions. Firstly, almost sure asymptotic stability is concerned (i) for regime-switching diffusions with finite state spaces by the Perron-Frobenius theorem, and, with regard to the case of reversible Markov chain, via the principal eigenvalue approach; (ii) for regime-switching diffusions with countable state spaces by means of a finite partition trick and an M-Matrix theory. We then apply our theory to study the stabilization for linear switching models. Several examples are given to demonstrate our theory.