Sharp non-asymptotic concentration inequalities for the approximation of the invariant distribution of a diffusion

Abstract

For an ergodic Brownian diffusion with invariant measure ν, we consider a sequence of empirical distributions (νn) n≥1 associated with an approximation scheme with decreasing time step (γn) n≥1 along an adapted regular enough class of test functions f such that f −ν(f ) is a coboundary of the infinitesimal generator A. Denote by σ the diffusion coefficient and ϕ the solution of the Poisson equation Aϕ = f − ν(f ). When the square norm |σ * ∇ϕ| 2 lies in the same coboundary class as f , we establish sharp non-asymptotic concentration bounds for suitable normalizations of νn(f ) − ν(f ). Our bounds are optimal in the sense that they match the asymptotic limit obtained by Lamberton and Pagès in [LP02], for a certain large deviation regime. In particular, this allows us to derive sharp non-asymptotic confidence intervals. Eventually, we are able to handle, up to an additional constraint on the time steps, Lipschitz sources f in an appropriate non-degenerate setting.

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

“…Remark that, in [5], a non-asymptotic Gaussian concentration was established with the asymptotically best constants for a particular large deviation called "Gaussian deviations" therein. In other words, for a = o( √ Γ n ):…”

“…To do so, we will perform the so-called martingales increments method which was exploited successfully by Frikha and Menozzi [3]. It was also the backbone of the analysis in [4] and [5]. Here, we adapt their techniques for the stochastic differential equation (E) driven by the compound Poisson with jump sizes satisfying Gaussian concentration.…”

“…Our techniques do not provide the sharp constant as in [5], this restriction comes from the deteriorating constants (depending on the dimension r) in the quasi Gaussian concentration property of the jump innovation (Z n ) n≥1 stated in Proposition 1.5. In other words, we cannot expect from our techniques to restore sharpness thanks to a suitable annex Poisson equation as in [5].…”

“…In the same way as the questions of the convergence of the empirical measure ν n or of its limiting distribution, the natural question is that of the nature of the deviations ν n (f ) − ν(f ) along appropriate test functions f . In the case of the Brownian diffusion this question was considered in [4] and [5]. Note that in the Brownian diffusion case the innovations of the Euler scheme are designed in order to "mimic" Brownian increments, hence it is natural to assume that they satisfy some Gaussian Concentration property (assumption (GC) in Sect.…”

“…(2.5)). An expansion of this kind is standard in this context, and analogous decompositions were already used in [4], [5] in continuous setting and [11], [10] with a jump component. It can be viewed as a kind of Itô formula for Euler scheme, because it permits to write the difference ϕ(X n ) − ϕ(X 0 ) as a sum of a martingale, a term involving the generator and a remainder term.…”