Exponential ergodicity of the jump-diffusion CIR process

TL;DR

This paper investigates the exponential ergodicity of the jump-diffusion CIR process, extending the classical model with jumps driven by a Lévy process, and establishes conditions for its ergodic behavior.

Contribution

It provides new conditions under which the jump-diffusion CIR process exhibits exponential ergodicity, including bounds on transition densities and Lyapunov functions.

Findings

Derived lower bounds for transition densities.

Established sufficient conditions for exponential ergodicity.

Proved existence of a Foster-Lyapunov function for JCIR.

Abstract

In this paper we study the jump-diffusion CIR process (shorted as JCIR), which is an extension of the classical CIR model. The jumps of the JCIR are introduced with the help of a pure-jump L\'evy process $(J_t, t \ge 0)$. Under some suitable conditions on the L\'evy measure of $(J_t, t \ge 0)$, we derive a lower bound for the transition densities of the JCIR process. We also find some sufficient condition guaranteeing the existence of a Forster-Lyapunov function for the JCIR process, which allows us to prove its exponential ergodicity.