Issue |
ESAIM: PS
Volume 22, 2018
|
|
---|---|---|
Page(s) | 58 - 95 | |
DOI | https://doi.org/10.1051/ps/2018009 | |
Published online | 11 October 2018 |
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
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.
Mathematics Subject Classification: 60G51 / 60G52 / 60H07 / 60H20 / 60H10 / 60J75
Key words: Lévy process / density in small time / stable process / Malliavin calculus for jump processes
© EDP Sciences, SMAI 2018
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