Nonparametric inference for discretely sampled Lévy processes

Given a sample from a discretely observed L\'evy process $X=(X_t)_{t\geq 0}$ of the finite jump activity, we study the problem of nonparametric estimation of the L\'evy density $\rho$ corresponding to the process $X.$ Our estimator of $\rho$ is based on a suitable inversion of the L\'evy-Khintchine formula and a plug-in device. The main result of the paper deals with an upper bound on the mean square error of the estimator of $\rho$ at a fixed point $x.$ We also show that the estimator attains the minimax convergence rate over a suitable class of L\'evy densities.