Nonparametric inference for discretely sampled Lévy processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 1, pp. 282-307.

Soit un échantillon d'un processus de Lévy X = (Xt)t≥0 à activité finie observé en temps discret, le problème d'estimation non-paramétrique de la densité de Lévy ρ est étudié. Un estimateur de ρ est proposé basé sur une inversion de Fourier de la formule de Lévy-Khintchine et un principe de plug-in. Les principaux résultats de cet article portent sur la majoration du risque de l'estimateur de ρ pour des classes de triplets de Lévy. La minoration du risque est aussi discutée.

Given a sample from a discretely observed Lévy process X = (Xt)t≥0 of the finite jump activity, the problem of nonparametric estimation of the Lévy density ρ corresponding to the process X is studied. An estimator of ρ is proposed that is based on a suitable inversion of the Lévy-Khintchine formula and a plug-in device. The main results of the paper deal with upper risk bounds for estimation of ρ over suitable classes of Lévy triplets. The corresponding lower bounds are also discussed.

DOI : 10.1214/11-AIHP433
Classification : 62G07, 62G20
Mots-clés : empirical characteristic function, empirical process, Fourier inversion, Lévy density, Lévy process, maximal inequality, mean square error 
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Gugushvili, Shota. Nonparametric inference for discretely sampled Lévy processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 1, pp. 282-307. doi : 10.1214/11-AIHP433. http://www.numdam.org/articles/10.1214/11-AIHP433/

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