Non-asymptotic concentration inequality for an approximation of the invariant distribution of a diffusion driven by compound poisson process.

Summary 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, particularly suitable in cases where the driving Lévy process is a Compound Poisson. This scheme is similar to those introduced by Lamberton and Pagès in [LP02] for a Brownian diffusion and extended by Panloup in [Pan08b] to the Jump Diffusion with Lévy jumps. We obtain a non-asymptotic Gaussian concentration bound for the difference between the invariant distribution and the empirical distribution computed with the scheme of decreasing time step along a appropriate test functions f such that f − ν(f) is is a coboundary of the infinitesimal generator.