Geodesic
Asymptotics in small time for the density of a stochastic differential equation driven by a stable Lévy process
ESAIM: Probability and Statistics, Tome 22 (2018), pp. 58-95

Voir la notice de l'article provenant de la source Numdam

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) 2316-1352.] 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. upper boundupper bound

DOI : 10.1051/ps/2018009
Classification : 60G51, 60G52, 60H07, 60H20, 60H10, 60J75
Keywords: Lévy process, density in small time, stable process, Malliavin calculus for jump processes

Clément, Emmanuelle 1 ; Gloter, Arnaud 1 ; Nguyen, Huong 1

1 
@article{PS_2018__22__58_0,
     author = {Cl\'ement, Emmanuelle and Gloter, Arnaud and Nguyen, Huong},
     title = {Asymptotics in small time for the density of a stochastic differential equation driven by a stable {L\'evy} process},
     journal = {ESAIM: Probability and Statistics},
     pages = {58--95},
     publisher = {EDP-Sciences},
     volume = {22},
     year = {2018},
     doi = {10.1051/ps/2018009},
     mrnumber = {3872128},
     zbl = {1405.60059},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ps/2018009/}
}
TY  - JOUR
AU  - Clément, Emmanuelle
AU  - Gloter, Arnaud
AU  - Nguyen, Huong
TI  - Asymptotics in small time for the density of a stochastic differential equation driven by a stable Lévy process
JO  - ESAIM: Probability and Statistics
PY  - 2018
SP  - 58
EP  - 95
VL  - 22
PB  - EDP-Sciences
UR  - http://geodesic.mathdoc.fr/articles/10.1051/ps/2018009/
DO  - 10.1051/ps/2018009
LA  - en
ID  - PS_2018__22__58_0
ER  - 
%0 Journal Article
%A Clément, Emmanuelle
%A Gloter, Arnaud
%A Nguyen, Huong
%T Asymptotics in small time for the density of a stochastic differential equation driven by a stable Lévy process
%J ESAIM: Probability and Statistics
%D 2018
%P 58-95
%V 22
%I EDP-Sciences
%U http://geodesic.mathdoc.fr/articles/10.1051/ps/2018009/
%R 10.1051/ps/2018009
%G en
%F PS_2018__22__58_0
Clément, Emmanuelle; Gloter, Arnaud; Nguyen, Huong. Asymptotics in small time for the density of a stochastic differential equation driven by a stable Lévy process. ESAIM: Probability and Statistics, Tome 22 (2018), pp. 58-95. doi: 10.1051/ps/2018009

Cité par Sources :