Regime-Switching Jump Diffusions with Non-Lipschitz Coefficients and Countably Many Switching States: Existence and Uniqueness, Feller, and Strong Feller Properties

TL;DR

This paper studies regime-switching jump diffusions with countably many states and non-Lipschitz coefficients, establishing existence, uniqueness, and Feller properties of solutions, advancing understanding of complex stochastic systems.

Contribution

It proves the existence and uniqueness of strong solutions for non-Lipschitz regime-switching jump diffusions with countably many states, and analyzes their Feller properties.

Findings

Unique strong solutions exist for the considered SDEs.

Feller and strong Feller properties are established.

The results extend the theory to non-Lipschitz coefficients and infinite switching states.

Abstract

This work focuses on a class of regime-switching jump diffusion processes, which is a two component Markov processes $(X(t),\Lambda(t))$, where $\Lambda(t)$ is a component representing discrete events taking values in a countably infinite set. Considering the corresponding stochastic differential equations, our main focus is on treating those with non-Lipschitz coefficients. We first show that there exists a unique strong solution to the corresponding stochastic differential equation. Then Feller and strong Feller properties are investigated.