Nonparametric inference for discretely sampled Lévy processes

Given a sample from a discretely observed Lévy process X = (Xt)t≥0 of the finite jump activity, the problem of nonparametric estimation of the Lévy density ρ corresponding to the process X is studied. An estimator of ρ is proposed that is based on a suitable inversion of the Lévy–Khintchine formula and a plug-in device. The main results of the paper deal with upper risk bounds for estimation of ρ over suitable classes of Lévy triplets. The corresponding lower bounds are also discussed