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

Emmanuelle Clément (1, 2) , Arnaud Gloter (3) , Huong Nguyen (4)

1 MICS - Mathématiques et Informatique pour la Complexité et les Systèmes
2 FR3487 - Fédération de Mathématiques de l'Ecole Centrale Paris
3 LaMME - Laboratoire de Mathématiques et Modélisation d'Evry
4 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

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.

Keywords: Lévy process; density in small time; stable process; Malliavin calculus for jump processes

Domains: Mathematics [math]

Emmanuelle Clément, Arnaud Gloter, Huong Nguyen. Asymptotics in small time for the density of a stochastic differential equation driven by a stable Lévy process. ESAIM: Probability and Statistics, 2018, 22, pp.58-95. ⟨10.1051/ps/2018009⟩. ⟨hal-01772290⟩