A Malliavin calculus method to study densities of additive functionals of SDE's with irregular drifts

Kohatsu-Higa, Arturo; Tanaka, Akihiro

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 3, pp. 871-883.

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.

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