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

In this article, we approximate the invariant distributionνof 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 PagèsBernoulli8(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-asymptoticquasiGaussian (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 functionsfsuch thatf−ν(f) is a coboundary of the infinitesimal generator.