Source: Ann. Inst. H. Poincaré Probab. Statist. Volume 46, Number 3
(2010), 595-617.
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.
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 adaptatatif 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.
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References
[1] O. E. Barndorff-Nielsen and N. Shephard. Modelling by Lévy processes for financial econometrics. In Lévy Processes. Theory and Applications 283–318. O. E. Barndorff-Nielsen, T. Mikosch and S. L. Resnick (Eds). Birkhauser, Boston, 2001.
[2] I. V. Basawa and P. J. Brockwell. Nonparametric estimation for nondecreasing Lévy processes. J. Roy. Statist. Soc. Ser. B 44 (1982) 262–269.
Mathematical Reviews (MathSciNet):
MR676217
[3] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996.
[4] L. Birgé and P. Massart. Minimum contrast estimators on sieves: Exponential bounds and rates of convergence. Bernoulli 4 (1998) 329–375.
[6] F. Comte, Y. Rozenholc and M.-L. Taupin. Penalized contrast estimator for adaptive density deconvolution. Canad. J. Statist. 34 (2006) 431–452.
[7] R. Cont and P. Tankov. Financial modelling with jump processes. In Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, 2004.
[8] P. J. Diggle and P. Hall. A Fourier approach to nonparametric deconvolution of a density estimate. J. Roy. Statist. Soc. Ser. B 55 (1993) 523–531.
[9] E. Eberlein and U. Keller. Hyperbolic distributions in finance. Bernoulli 1 (1995) 281–299.
[10] J. Fan. On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 (1991) 1257–1272.
[11] J. E. Figueroa-López and C. Houdré. Risk bounds for the nonparametric estimation of Lévy processes. IMS Lecture Notes Monogr. Ser. 51 (2006) 96–116.
[12] G. Jongbloed and F. H. van der Meulen. Parametric estimation for subordinators and induced OU processes. Scand. J. Statist. 33 (2006) 825–847.
[13] G. Jongbloed, F. H. van der Meulen and A. W. van der Vaart. Nonparametric inference for Lévy-driven Ornstein–Uhlenbeck processes. Bernoulli 11 (2005) 759–791.
[14] T. Klein and E. Rio. Concentration around the mean for maxima of empirical processes. Ann. Probab. 33 (2005) 1060–1077.
[15] U. Küchler and S. Tappe. Bilateral Gamma distributions and processes in financial mathematics. Stochastic Processes Appl. 118 (2008) 261–283.
[16] M. Ledoux and M. Talagrand. Probability in Banach Spaces. Isoperimetry and Processes. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge 23. Springer, Berlin, 1991.
[17] D. B. Madan and E. Seneta. The Variance Gamma (V. G.) model for share market returns. The Journal of Business 63 (1990) 511–524.
[18] Y. Meyer. Ondelettes et opérateurs I. Hermann, Paris, 1990.
[19] M. Neumann. On the effect of estimating the error density in nonparametric deconvolution. J. Nonparametr. Stat. 7 (1997) 307–330.
[20] M. Neumann and M. Reiss. Nonparametric estimation for Lévy processes from low-frequency observations. Bernoulli 15 (2009) 223–248.
[21] K. I. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge, 1999.
[22] R. N. Watteel and R. J. Kulperger. Nonparametric estimation of the canonical measure for infinitely divisible distributions. J. Stat. Comput. Simul. 73 (2003) 525–542.
