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Adaptive kernel estimation of the Lévy density
Mélina Bec1, Claire Lacour2

This paper is concerned with adaptive kernel estimation of the Lévy density $N(x)$ for pure jump Lévy 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.
1 :  MAP5 - Mathématiques appliquées Paris 5
2 :  LM-Orsay - Laboratoire de Mathématiques d'Orsay