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 nn discrete time observations with step ΔΔ. The asymptotic framework is: nn tends to infinity, Δ=ΔnΔ=Δn tends to zero while nΔnnΔn 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 L2L2-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 L2L2-risk restricted to the compact interval. We discuss rates of convergence and give examples and simulation results for processes fitting in our framework.