On introduit une méthode générale qui permet l'utilisation du Calcul de Malliavin pour des fonctionnelles additives générées par des équations stochastiques avec une dérive irrégulière. Cette méthode utilise le théorème de Girsanov avec l'expansion d'Itô-Taylor pour obtenir la régularité de la densité. On applique cette méthodologie pour au cas de l'intégrale en temps d'une diffusion avec derive mesurable bornée.
We present a general method which allows to use Malliavin Calculus for additive functionals of stochastic equations with irregular drift. This method uses the Girsanov theorem combined with Itô-Taylor expansion in order to obtain regularity properties for this density. We apply the methodology to the case of the Lebesgue integral of a diffusion with bounded and measurable drift.
Mots clés : Malliavin calculus, non-smooth drift, density function
@article{AIHPB_2012__48_3_871_0, author = {Kohatsu-Higa, Arturo and Tanaka, Akihiro}, title = {A {Malliavin} calculus method to study densities of additive functionals of {SDE's} with irregular drifts}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {871--883}, publisher = {Gauthier-Villars}, volume = {48}, number = {3}, year = {2012}, doi = {10.1214/11-AIHP418}, mrnumber = {2976567}, zbl = {1248.60058}, language = {en}, url = {http://www.numdam.org/articles/10.1214/11-AIHP418/} }
TY - JOUR AU - Kohatsu-Higa, Arturo AU - Tanaka, Akihiro TI - A Malliavin calculus method to study densities of additive functionals of SDE's with irregular drifts JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2012 SP - 871 EP - 883 VL - 48 IS - 3 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/11-AIHP418/ DO - 10.1214/11-AIHP418 LA - en ID - AIHPB_2012__48_3_871_0 ER -
%0 Journal Article %A Kohatsu-Higa, Arturo %A Tanaka, Akihiro %T A Malliavin calculus method to study densities of additive functionals of SDE's with irregular drifts %J Annales de l'I.H.P. Probabilités et statistiques %D 2012 %P 871-883 %V 48 %N 3 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/11-AIHP418/ %R 10.1214/11-AIHP418 %G en %F AIHPB_2012__48_3_871_0
Kohatsu-Higa, Arturo; Tanaka, Akihiro. A Malliavin calculus method to study densities of additive functionals of SDE's with irregular drifts. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 3, pp. 871-883. doi : 10.1214/11-AIHP418. http://www.numdam.org/articles/10.1214/11-AIHP418/
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