Extensions of the Itô formula through Malliavin calculus and application to a variational problem

Abstract : This dissertation studies the extension of the Itô formula to the case of distibution-valued paths of bounded variation lifted by processes which are regular in the sense of Malliavin calculus. We make optimal hypotheses, which gives us access to many applications. The first chapter is a primer in Malliavin calculus. The second chapter provides useful results on the toplogy of the schwartz class and of the space of tempered distributions. in the third chapter, we give optimal conditions under which a tempered distribution may be composed by a random variable and we study the malliavin regularity of the object thus defined. Interpolation techniques give access to results in fractional spaces. We also give results for the case where the tempered distribution is itself stochastic. These results allow us to obtain, in chapter 4, a weak Itô formula under hypotheses which are much weaker than those usually made in the litterature. We also give an Itô-Wentzell and an anticipative version. In the case where the process to which the ito formula is applied is the solution to an SDE, we give a more precise result, which we use to study the reguarity of the multi-dimensional local time. Finally the fifth chapter solves a variational problem under hypotheses which are much weaker than the usual assumption of hypoellipticity
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Jérôme Valentin. Extensions of the Itô formula through Malliavin calculus and application to a variational problem. General Mathematics [math.GM]. Télécom ParisTech, 2012. English. ⟨NNT : 2012ENST0029⟩. ⟨tel-01136570⟩

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