Adaptive kernel estimation of the Lévy density

This paper is concerned with adaptive kernel estimation of the Levy density $N(x)$ for pure jump Levy processes. The sample path is observed at $n$ discrete instants in the "high frequency" context ($ \Delta $ = $ \Delta_n $ tends to zero while $n \Delta_n $ 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.