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

Abstract

This work focuses on the asymptotic behavior of the density in small time of a stochastic differential equation driven by a truncatedα-stable process with indexα∈ (0, 2). We assume that the process depends on a parameterβ= (θ,σ)Tand 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 indexα∈ (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.