Comparison theorem and stability under perturbation of transition rate matrices for regime-switching processes

TL;DR

This paper develops a comparison theorem and provides stability estimates for regime-switching diffusion processes, allowing better control and understanding of their behavior under perturbations of transition rates, applicable even in infinite state spaces.

Contribution

It introduces a novel comparison theorem and sharp stability estimates for state-dependent regime-switching processes, extending analysis to infinite state spaces and non-birth-death types.

Findings

Established a comparison theorem controlling switching process evolution.

Provided sharp stability estimates under transition rate perturbations.

Improved results on ergodicity and stability of regime-switching processes.

Abstract

A comparison theorem for state-dependent regime-switching diffusion processes is established, which enables us to control pathwisely the evolution of the state-dependent switching component simply by Markov chains. Moreover, a sharp estimate on the stability of Markovian regime-switching processes under the perturbation of transition rate matrices is provided. Our approach is based on the elaborate constructions of switching processes in the spirit of Skorokhod's representation theorem varying according to the problem being dealt with. In particular, this method can cope with the switching processes in an infinite state space and not necessarily being of birth-death type. As an application, some known results on ergodicity and stability of state-dependent regime-switching processes can be improved.