Nonparametric inference for discretely sampled Lévy processes

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