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 Δ . The asymptotic framework is: n tends to infinity, Δ = Δ n tends to zero while n Δ 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 L 2 -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 L 2 -risk restricted to the compact interval. We discuss rates of convergence and give examples and simulation results for processes fitting in our framework.