Adaptive pointwise estimation for pure jump Lévy processes

This paper is concerned with adaptive kernel estimation of the L\évy density N(x) for bounded-variation pure-jump L\évy processes. The sample path is observed at n discrete instants in the "high frequency" context (\Delta = \Delta(n) tends to zero while n\Delta tends to infinity). We construct a collection of kernel estimators of the function g(x)=xN(x) and propose a method of local adaptive selection of the bandwidth. We provide an oracle inequality and a rate of convergence for the quadratic pointwise risk. This rate is proved to be the optimal minimax rate. We give examples and simulation results for processes fitting in our framework. We also consider the case of irregular sampling.