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

Dawid Czapla, Joanna Kubieniec

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.

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.

Paper Structure

This paper contains 9 sections, 4 theorems, 129 equations.

Table of Contents

  1. Preliminaries
  2. A general criterion on the exponential ergodicity for Markov--Feller operators
  3. Exponential ergodicity for some random dynamical system
  4. Model description and assumptions
  5. Exponential ergodicity of the jump opertor
  6. Application to a Poisson-driven stochastic differential equation
  7. Poisson random measure and Poisson point process
  8. Model description and assumptions
  9. Exponential ergodicity of the jump operator associated with the PDSDE

Key Result

Theorem 2.1

Suppose that $P:\mathcal{M}_{+}(E)\to \mathcal{M}_{+}(E)$ is a Markov operator, which enjoys the Feller property, and that there exists a substochastic kernel $Q$ on $E^2\times\mathcal{B}(E^2)$ satisfying qeq. Furthermore, assume that the following conditions hold: Then the operator $P$ possesses a unique invariant measure $\mu^{*}\in \mathcal{M}_{prob}(E)$ and $\left\langle V,\mu^*\right\rangle<

Theorems & Definitions (6)

  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • proof