Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A Malliavin Calculus method to study densities of additive functionals of SDE’s with irregular drifts

Arturo Kohatsu-Higa and Akihiro Tanaka

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 3 (2012), 871-883.

Dates
First available in Project Euclid: 26 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1340714876

Digital Object Identifier
doi:10.1214/11-AIHP418

Mathematical Reviews number (MathSciNet)
MR2976567

Zentralblatt MATH identifier
1248.60058

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
Malliavin Calculus Non-smooth drift Density function

Citation

Kohatsu-Higa, Arturo; Tanaka, Akihiro. A Malliavin Calculus method to study densities of additive functionals of SDE’s with irregular drifts. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 3, 871--883. doi:10.1214/11-AIHP418. https://projecteuclid.org/euclid.aihp/1340714876


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