Stability and recurrence of regime-switching diffusion processes

TL;DR

This paper establishes criteria for the stability and recurrence of regime-switching diffusion processes, introducing new methods applicable to both finite and infinite state spaces, including nonlinear cases.

Contribution

It presents two novel methods for analyzing stability and recurrence in regime-switching diffusions, especially in infinite state spaces, using M-matrix theory and principal eigenvalues.

Findings

Provided criteria for stability of regime-switching diffusions.

Developed two methods for infinite state space analysis.

Applied principal eigenvalue approach to recurrence studies.

Abstract

We provide some criteria on the stability of regime-switching diffusion processes. Both the state-independent and state-dependent regime-switching diffusion processes with switching in a finite state space and an infinite countable state space are studied in this work. We provide two methods to deal with switching processes in an infinite countable state space. One is a finite partition method based on the nonsingular M-matrix theory. Another is an application of principal eigenvalue of a bilinear form. Our methods can deal with both linear and nonlinear regime-switching diffusion processes. Moreover, the method of principal eigenvalue is also used to study the recurrence of regime-switching diffusion processes.