Regime-switching diffusion processes: strong solutions and strong Feller property

TL;DR

This paper studies regime-switching diffusion processes, establishing conditions for strong solutions, non-explosion, and the strong Feller property, under general assumptions on the diffusion components.

Contribution

It provides new existence, uniqueness, and non-explosion criteria for regime-switching diffusions without restrictive coefficient conditions.

Findings

Existence and uniqueness of strong solutions up to explosion time.

Non-explosion conditions for regime-switching diffusions.

Proof of the strong Feller property under certain conditions.

Abstract

We investigate the existence and uniqueness of strong solutions up to an explosion time for regime-switching diffusion processes in an infinite state space. Instead of concrete conditions on coefficients, our existence and uniqueness result is established under the general assumption that the diffusion in every fixed environment has a unique non-explosive strong solution. Moreover, non-explosion conditions for regime-switching diffusion processes are given. The strong Feller property is proved by further assuming that the diffusion in every fixed environment generates a strong Feller semigroup.