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Approximation of the invariant distribution for a class of ergodic jump diffusions. (English) Zbl 1455.60091

Summary: In this article, we approximate the invariant distribution \(\nu\) 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 [D. Lamberton and G. Pagès, Bernoulli 8, No. 3, 367–405 (2002; Zbl 1006.60074)] for a Brownian diffusion and extended in [F. Panloup, Ann. Appl. Probab. 18, No. 2, 379–426 (2008; Zbl 1136.60049)] 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 \(f-\nu (f)\) is a coboundary of the infinitesimal generator.

MSC:

60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60G51 Processes with independent increments; Lévy processes
60E15 Inequalities; stochastic orderings
65C30 Numerical solutions to stochastic differential and integral equations
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