Asymptotics in small time for the density of a stochastic differential equation driven by a stable Lévy process

¹ Laboratoire MICS, Fédération de Mathématiques FR 3487, Université Paris-Saclay, CentraleSupélec, 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: 12 December 2016
Accepted: 5 March 2018

Abstract

This work focuses on the asymptotic behavior of the density in small time of a stochastic differential equation driven by a truncated α-stable process with index α ∈ (0, 2). We assume that the process depends on a parameter β = (θ, σ)^T and we study the sensitivity of the density with respect to this parameter. This extends the results of [E. Clément and A. Gloter, Local asymptotic mixed normality property for discretely observed stochastic dierential equations driven by stable Lévy processes. Stochastic Process. Appl. 125 (2015) 2316–2352.] which was restricted to the index α ∈ (1, 2) and considered only the sensitivity with respect to the drift coefficient. By using Malliavin calculus, we obtain the representation of the density and its derivative as an expectation and a conditional expectation. This permits to analyze the asymptotic behavior in small time of the density, using the time rescaling property of the stable process.