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

Abstract

This work focuses on a class of regime-switching jump diffusion processes, which is a two component Markov processes (X(t),Λ (t)), where Λ (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.

On Strong Feller Property, Exponential Ergodicity and Large Deviations Principle for Stochastic Damping Hamiltonian Systems with State-Dependent Switching

“…We can use Theorem 2.5 of Xi et al (2019) to verify that the system (6.32)-(6.33) has a unique non-explosive strong solution (X, Λ). Since the diffusion coefficient obviously satisfy the uniform ellipticity condition, the process (X, Λ) is strong Feller.…”

Periodic solutions of hybrid jump diffusion processes

“…Xi [13], Xi and Yin [14] investigated asymptotic properties of the model. Nguyen and Yin [10,11], Shao [12], Xi and Zhu [16], Xi, Yin and Zhu [15] considered the model whose switching component has a countably infinite state space. Note that most of the existing literatures have focused on the time-homogeneous case.…”

“…Different from [14, Proposition 2.1], we only assume that the coefficient functions satisfy the local Lipschitz condition. Also, we remove a key assumption of [12,15,16], which requires that the transition matrix of the switching component is Hölder continuous. In Section 3, we will establish the existence and uniqueness of periodic solutions.…”

Exponential ergodicity for regime-switching diffusion processes in total variation norm

“…where C 5 (p) is defined by (33). Letting R > R 0 be sufficiently large such that C 5 (p) > 0, we obtain…”

“…In particular, we would like to mention [22] in which with Lipschitz coefficients, Lyapunov condition and boundedness assumption, Shao studied exponential ergodicity in Wasserstein distance by virtue of coupling method developed by [5,6]. However, as is well-known, Lipschitz condition is somewhat restrictive in practical applications; see [33]. What is more, it is difficult to find a suitable Lyapunov function for the regime-switching diffusion processes; see [23].…”