LAMN property for the drift and volatility parameters of a sde driven by a stable Lévy process

¹ Mathématiques et Informatique pour la Complexité, Fédération de Mathématiques FR 3487, CentraleSupélec, Université Paris-Saclay, 91190 Gif-sur-Yvette, France.
² Université d’Évry Val d’Essonne, Laboratoire de Mathématiques et Modélisation d’Evry, UMR 8071, 23 Boulevard de France, 91037 Évry Cedex, France.
³ Université Paris-Est, LAMA UMR 8050, UPEMLV, CNRS, UPEC, 77454 Marne-la-Vallée, France.

^* Corresponding author: emmanuelle.clement@u-pem.fr

Received: 27 February 2017
Accepted: 24 January 2018

Abstract

This work focuses on the local asymptotic mixed normality (LAMN) property from high frequency observations, of a continuous time process solution of a stochastic differential equation driven by a truncated α-stable process with index α ∈ (0, 2). The process is observed on the fixed time interval [0,1] and the parameters appear in both the drift coefficient and scale coefficient. This extends the results of Clément and Gloter [Stoch. Process. Appl. 125 (2015) 2316–2352] where the index α ∈ (1, 2) and the parameter appears only in the drift coefficient. We compute the asymptotic Fisher information and find that the rate in the LAMN property depends on the behavior of the Lévy measure near zero. The proof relies on the small time asymptotic behavior of the transition density of the process obtained in Clément et al. [Preprint HAL-01410989v2 (2017)].