Nonparametric adaptive estimation for pure jump Lévy processes

Abstract. Motivated by Neumann and Reiss (2007), this paper is concerned with non-parametric estimation of the Lévy density of a pure jump Lévy process. The sample path is observed at n discrete instants with fixed sampling interval. We construct a collection of estimators obtained by deconvolution methods and deduced from appropriate estima-tors of the characteristic function and its first derivative. The deconvolution methods are inspired by (Comte et al, 2006). We obtain a bound for the L2-risk, under general assumptions on the model. Then we propose a penalty function that allows to build an adaptive estimator. The risk bound for the adaptive estimator is obtained under addi-tional assumptions on the Lévy density. Examples of models fitting in our framework are described and rates of convergence of the estimator are discussed. References [1] Comte, F., Rozenholc, Y. and Taupin, M.-L. (2006) Penalized contrast estimator for adaptive density deconvolution. Canad. J. Statist. 34, 431-452. [2] Neumann, M. and Reiss, M. (2007). Nonparametric estimation for Lévy processes from low-frequenc