[Propriété LAN pour un processus de Lévy simple]
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.
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@article{CRMATH_2014__352_10_859_0,
author = {Kohatsu-Higa, Arturo and Nualart, Eulalia and Tran, Ngoc Khue},
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},
url = {http://www.numdam.org/articles/10.1016/j.crma.2014.08.013/}
}
TY - JOUR AU - Kohatsu-Higa, Arturo AU - Nualart, Eulalia AU - Tran, Ngoc Khue TI - LAN property for a simple Lévy process JO - Comptes Rendus. Mathématique PY - 2014 SP - 859 EP - 864 VL - 352 IS - 10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2014.08.013/ DO - 10.1016/j.crma.2014.08.013 LA - en ID - CRMATH_2014__352_10_859_0 ER -
%0 Journal Article %A Kohatsu-Higa, Arturo %A Nualart, Eulalia %A Tran, Ngoc Khue %T LAN property for a simple Lévy process %J Comptes Rendus. Mathématique %D 2014 %P 859-864 %V 352 %N 10 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2014.08.013/ %R 10.1016/j.crma.2014.08.013 %G en %F CRMATH_2014__352_10_859_0
Kohatsu-Higa, Arturo; Nualart, Eulalia; Tran, Ngoc Khue. LAN property for a simple Lévy process. Comptes Rendus. Mathématique, Tome 352 (2014) no. 10, pp. 859-864. doi : 10.1016/j.crma.2014.08.013. http://www.numdam.org/articles/10.1016/j.crma.2014.08.013/
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