Nonparametric inference for discretely sampled Lévy processes

Given a sample from a discretely observed Lévy process $ X = (X_t)_{t \geq 0} $ of the finite jump activity, the problem of nonparametric estimation of the Lévy density \rho corresponding to the process X is studied. An estimator of \rho 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 \rho over suitable classes of Lévy triplets. The corresponding lower bounds are also discussed.