Orateur: Emmanuelle Clément ( Professeur, laboratoire MICS, CentraleSupélec)
Titre: Density in small time for Lévy driven SDE and applications to parametric estimation
Modeling with pure jump processes has become very popular in the last decade, especially in energy markets. In this work, we focus on the parametric estimation of a stochastic differential equation driven by a stable Lévy process. We first study asymptotics for the transition density of the process in small time. By using Malliavin calculus, we obtain some representations of the density and its derivatives with respect to drift and scale parameters as expectations. This permits to analyze the asymptotic behavior in small time of the density, using the time rescaling property of the stable process. Then we apply these results to parametric estimation of drift and scale parameters of the process based on high frequency observations on a fixed time interval. We prove the Local Asymptotic Mixed Normality (LAMN) property and compute the asymptotic Fisher information as well as the optimal rate of convergence in estimating the parameters.