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

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.

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• The central limit theorem for Markov processes that are exponentially ergodic in the bounded-Lipschitz norm
• On the Existence and Uniqueness of Stationary Distributions for Some Piecewise Deterministic Markov Processes With State-Dependent Jump Intensity