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

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.

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.

DOI : https://doi.org/10.1214/11-AIHP433
Classification:  62G07,  62G20
Keywords: empirical characteristic function, empirical process, Fourier inversion, Lévy density, Lévy process, maximal inequality, mean square error
@article{AIHPB_2012__48_1_282_0,
     author = {Gugushvili, Shota},
     title = {Nonparametric inference for discretely sampled L\'evy processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {1},
     year = {2012},
     pages = {282-307},
     doi = {10.1214/11-AIHP433},
     zbl = {1235.62121},
     mrnumber = {2919207},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2012__48_1_282_0}
}
Gugushvili, Shota. Nonparametric inference for discretely sampled Lévy processes. Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 1, pp. 282-307. doi : 10.1214/11-AIHP433. http://www.numdam.org/item/AIHPB_2012__48_1_282_0/

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