Nonparametric estimation for pure jump Lévy processes based on high frequency data

In this paper, we study nonparametric estimation of the Lévy density for pure jump Lévy processes. We consider n discrete time observations with step [Delta]. The asymptotic framework is: n tends to infinity, [Delta]=[Delta]n tends to zero while n[Delta]n tends to infinity. First, we use a Fourier approach ("frequency domain"): this allows us to construct an adaptive nonparametric estimator and to provide a bound for the global -risk. Second, we use a direct approach ("time domain") which allows us to construct an estimator on a given compact interval. We provide a bound for -risk restricted to the compact interval. We discuss rates of convergence and give examples and simulation results for processes fitting in our framework.