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

Clément, Emmanuelle; Gloter, Arnaud; Nguyen, Huong

ESAIM: Probability and Statistics, Tome 22 (2018), pp. 58-95.

This work focuses on the asymptotic behavior of the density in small time of a stochastic differential equation driven by a truncated $\alpha$-stable process with index $\alpha \in (0,2)$. We assume that the process depends on a parameter $\beta ={(\theta ,\sigma )}^T$ 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) 2316-1352.] which was restricted to the index $\alpha \in (1,2)$ 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. upper boundupper bound

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