Exponential ergodicity of some Markov dynamical system with application to a Poisson driven stochastic differential equation

TL;DR

This paper investigates the long-term behavior of a class of Markov processes with state-dependent jumps, establishing conditions for exponential convergence to a unique invariant distribution and extending existing results to Poisson-driven stochastic differential equations.

Contribution

It provides new sufficient conditions for exponential ergodicity of certain Markov systems and generalizes previous results to more complex Poisson-driven SDEs with solution-dependent intensities.

Findings

Existence of a unique invariant distribution under specified conditions

Exponential convergence in the dual bounded Lipschitz distance

Extension of Kazak's result to broader Poisson-driven SDEs

Abstract

We are concerned with the asymptotics of the Markov chain given by the post-jump locations of a certain piecewise-deterministic Markov process with a state-dependent jump intensity. We provide sufficient conditions for such a model to possess a unique invariant distribution, which is exponentially attracting in the dual bounded Lipschitz distance. Having established this, we generalise a result of J. Kazak on the jump process defined by a Poisson driven stochastic differential equation with a solution-dependent intensity of perturbations.