Abstract
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.
Highlights
Σ, and κ satisfy a suitable Lyapunov condition (assumption (LV) in Sect. 1.3) which ensures the existence of an invariant distribution ν of (Xt)t≥0
When the diffusion contains Levy jumps, it is not generally expected that these deviations are Gaussian like which is not in accordance with the Central Limit Theorem (CLT) from [10]. Such a behavior seems natural if we suppose that the driving Levy process is a Compound Poisson process and the jump size vectors (Yk)k∈N satisfy a Gaussian Concentration property (GC)
(GC) We say that a random variable G ∈ L1 satisfies the Gaussian concentration property, if for every Lipschitz continuous function g : Rr → R and every λ > 0: E exp(λg(G)) ≤ exp λE
Summary
Note that in the Brownian diffusion case the innovations of the Euler scheme are designed in order to “mimic” Brownian increments, it is natural to assume that they satisfy some Gaussian Concentration property (assumption (GC) in Sect. When the diffusion contains Levy jumps, it is not generally expected that these deviations are Gaussian like which is not in accordance with the CLT from [10] Such a behavior seems natural if we suppose that the driving Levy process is a Compound Poisson process and the jump size vectors (Yk)k∈N satisfy a Gaussian Concentration property (GC).
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