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

Abstract. In this paper, we study nonparametric estimation of the Lévy density for pure jump Lévy processes. We consider n discrete time observations with step ∆. The asymptotic framework is: n tends to infinity, ∆ = ∆n tends to zero while n∆n tends to infinity. First, we use a Fourier approach (“frequency domain”): this allows to construct an adaptive nonparametric estimator and to provide a bound for the global L2-risk. Second, we use a direct approach (“time domain”) which allows to construct an estimator on a given compact interval. We provide a bound for L2-risk restricted to the compact interval. We discuss rates of convergence and give examples and simulation results for processes fitting in our framework. February 3, 2009 Keywords. Adaptive nonparametric estimation. High frequency data. Lévy processes. Projection estimators. AMS Classification. 62G05-62M05-60G51. 1