Approximation of the invariant distribution for a class of ergodic jump diffusions

Université d’Évry Val d’Essonne, Université Paris-Saclay, CNRS, Univ Evry, Laboratoire de Mathématiques et Modélisation d’Evry, 91037 Evry, France.

^* Corresponding author: igor.honore@inria.fr

Received: 19 April 2019
Accepted: 18 September 2020

Abstract

In this article, we approximate the invariant distribution ν of an ergodic Jump Diffusion driven by the sum of a Brownian motion and a Compound Poisson process with sub-Gaussian jumps. We first construct an Euler discretization scheme with decreasing time steps. This scheme is similar to those introduced in Lamberton and Pagès Bernoulli 8 (2002) 367-405. for a Brownian diffusion and extended in F. Panloup, Ann. Appl. Probab. 18 (2008) 379-426. to a diffusion with Lévy jumps. We obtain a non-asymptotic quasi Gaussian (asymptotically Gaussian) concentration bound for the difference between the invariant distribution and the empirical distribution computed with the scheme of decreasing time step along appropriate test functions f such that f − ν(f) is a coboundary of the infinitesimal generator.