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

摘要

In this paper, we study nonparametric estimation of the Levy density for pure jump Levy 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 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.