Comptes Rendus
no. 10
Probability theory/Statistics
LAN property for a simple Lévy process

Presented by: 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.

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.

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.

Received:
Accepted:
Published online:
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 
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     title = {LAN property for a simple {L\'evy} process},
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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/

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