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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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.


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

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

First available in Project Euclid: 26 June 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Malliavin Calculus Non-smooth drift Density function


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.

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