Nonparametric adaptive estimation for pure jump Lévy processes

Comte, F.; Genon-Catalot, V.

Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 3, pp. 595-617.

Ce travail étudie l'estimation non paramétrique de la densité d'un processus de Lévy de saut pur. Les trajectoires sont observées à n instants discrets de pas fixé. Nous construisons une collection d'estimateurs obtenus par des méthodes de type déconvolution, et s'appuyant sur des estimateurs pertinents de la fonction caractéristique et de ses dérivées. Sous des hypothèses assez générales sur le modèle, nous obtenons une borne pour le risque quadratique intégré. Nous proposons ensuite une pénalité permettant de construire un estimateur adaptatif. La borne de risque de l'estimateur adaptatif est obtenue sous des hypothèses supplémentaires sur la densité de la mesure de Lévy. Nous donnons pour finir des exemples de modèles adaptés à notre contexte et nous calculons dans chaque cas la vitesse de convergence de l'estimateur.

This paper is concerned with nonparametric 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 estimators of the characteristic function and its first derivative. We obtain a bound for the -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 additional assumptions on the Lévy density. Examples of models fitting in our framework are described and rates of convergence of the estimator are discussed.

Classification : 62G05, 62M05, 60G51
Mots clés : adaptive estimation, deconvolution, Lévy process, nonparametric projection estimator

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