Stability of Regime-Switching Diffusion Systems with Discrete States Belonging to a Countable Set

TL;DR

This paper investigates the stability of regime-switching diffusion systems with a countably infinite discrete state space, providing new practical conditions and estimates for the rate of convergence to stability.

Contribution

It introduces a feasible stability analysis approach for systems with infinite discrete states and derives path-wise convergence rates, extending classical Lyapunov methods.

Findings

Established stability conditions under practical assumptions

Derived estimates for the convergence rate to stability

Provided illustrative examples demonstrating the results

Abstract

This work focuses on stability of regime-switching diffusions consisting of continuous and discrete components, in which the discrete component switches in a countably infinite set and its switching rates at current time depend on the continuous component. In contrast to the existing approach, this work provides more practically viable approach with more feasible conditions for stability. A classical approach for asymptotic stabilityusing Lyapunov function techniques shows the Lyapunov function evaluated at the solution process goes to 0 as time $t\to \infty$. A distinctive feature of this paper is to obtain estimates of path-wise rates of convergence, which pinpoints how fast the aforementioned convergence to 0 taking place. Finally, some examples are given to illustrate our findings.