Comptes Rendus
no. 10
Probability theory/Statistics
LAN property for a simple Lévy process
[Propriété LAN pour un processus de Lévy simple]

Présenté par : the Editorial Board

Arturo Kohatsu-Higa1 ; Eulalia Nualart2 ; Ngoc Khue Tran3
1 Department of Mathematical Sciences, Ritsumeikan University and Japan Science and Technology Agency, 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan
2 Dept. Economics and Business, Universitat Pompeu Fabra and Barcelona Graduate School of Economics, Ramón Trias Fargas 25-27, 08005 Barcelona, Spain
3 Université Paris-13, Sorbonne Paris Cité, LAGA, CNRS, UMR 7539, 93430 Villetaneuse, France
Comptes Rendus. Mathématique, Volume 352 (2014) no. 10, pp. 859-864.

Dans cet article, nous considérons un processus de Lévy simple donné par un mouvement brownien et un processus de Poisson compensé, dont les paramètres et l'intensité sont inconnus. En supposant que le processus est observé à haute fréquence, nous obtenons la propriété de normalité asymptotique locale. Pour cela, le calcul de Malliavin et le théorème de Girsanov sont appliqués afin d'écrire le logarithme du rapport de vraisemblances comme une somme d'espérances conditionnelles, pour laquelle un théorème central limite pour des suites triangulaires peut être appliqué.

In this paper, we consider a simple Lévy process given by a Brownian motion and a compensated Poisson process, whose drift and diffusion parameters as well as its intensity are unknown. Supposing that the process is observed discretely at high frequency, we derive the local asymptotic normality (LAN) property. In order to obtain this result, Malliavin calculus and Girsanov's theorem are applied in order to write the log-likelihood ratio in terms of sums of conditional expectations, for which a central limit theorem for triangular arrays can be applied.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.08.013
Arturo Kohatsu-Higa 1 ; Eulalia Nualart 2 ; Ngoc Khue Tran 3

1 Department of Mathematical Sciences, Ritsumeikan University and Japan Science and Technology Agency, 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan
2 Dept. Economics and Business, Universitat Pompeu Fabra and Barcelona Graduate School of Economics, Ramón Trias Fargas 25-27, 08005 Barcelona, Spain
3 Université Paris-13, Sorbonne Paris Cité, LAGA, CNRS, UMR 7539, 93430 Villetaneuse, France 
@article{CRMATH_2014__352_10_859_0,
     author = {Arturo Kohatsu-Higa and Eulalia Nualart and Ngoc Khue Tran},
     title = {LAN property for a simple {L\'evy} process},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {859--864},
     publisher = {Elsevier},
     volume = {352},
     number = {10},
     year = {2014},
     doi = {10.1016/j.crma.2014.08.013},
     language = {en},
}
TY  - JOUR
AU  - Arturo Kohatsu-Higa
AU  - Eulalia Nualart
AU  - Ngoc Khue Tran
TI  - LAN property for a simple Lévy process
JO  - Comptes Rendus. Mathématique
PY  - 2014
SP  - 859
EP  - 864
VL  - 352
IS  - 10
PB  - Elsevier
DO  - 10.1016/j.crma.2014.08.013
LA  - en
ID  - CRMATH_2014__352_10_859_0
ER  - 
%0 Journal Article
%A Arturo Kohatsu-Higa
%A Eulalia Nualart
%A Ngoc Khue Tran
%T LAN property for a simple Lévy process
%J Comptes Rendus. Mathématique
%D 2014
%P 859-864
%V 352
%N 10
%I Elsevier
%R 10.1016/j.crma.2014.08.013
%G en
%F CRMATH_2014__352_10_859_0
Arturo Kohatsu-Higa; Eulalia Nualart; Ngoc Khue Tran. LAN property for a simple Lévy process. Comptes Rendus. Mathématique, Volume 352 (2014) no. 10, pp. 859-864. doi : 10.1016/j.crma.2014.08.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.08.013/

[1] Y. Aït-Sahalia; J. Jacod Fisher's information for discretely sampled Lévy processes, Econometrica, Volume 76 (2008), pp. 727-761

[2] E. Clément; S. Delattre; A. Gloter Asymptotic lower bounds in estimating jumps, Bernoulli, Volume 20 (2014), pp. 1059-1096

[3] V. Genon-Catalot; J. Jacod On the estimation of the diffusion coefficient for multi-dimensional diffusion processes, Ann. Inst. Henri Poincaré B, Probab. Stat., Volume 29 (1993), pp. 119-151

[4] A. Gloter; E. Gobet LAMN property for hidden processes: the case of integrated diffusions, Ann. Inst. Henri Poincaré B, Probab. Stat., Volume 44 (2008), pp. 104-128

[5] E. Gobet LAMN property for elliptic diffusions: a Malliavin calculus approach, Bernoulli, Volume 7 (2001), pp. 899-912

[6] E. Gobet LAN property for ergodic diffusions with discrete observations, Ann. Inst. Henri Poincaré, Volume 38 (2002), pp. 711-737

[7] R. Kawai LAN property for Ornstein–Uhlenbeck processes with jumps under discrete sampling, J. Theor. Probab., Volume 26 (2013), pp. 932-967

[8] T. Ogihara; N. Yoshida Quasi-likelihood analysis for the stochastic differential equation with jumps, Stat. Inference Stoch. Process., Volume 14 (2011), pp. 189-229

Cité par Sources :

Commentaires - Politique