Basics of Malliavin Calculus

Abstract

[en] This work is an introduction to Malliavin calculus. We start by giving the definition of an integration by parts formula and how they are related to the existence of densities of random variables. The central topic of this work is how using Malliavin calculus we can find integration by parts formulas. In order to accomplish this objective, there are presented tools such as the Wiener chaos decomposition, the multiple Wiener-Itô integral and the fundamental operators which are: the differential operator, the divergence operator and the generator of the Ornstein–Uhlenbeck semigroup. These operators are combined to obtain explicit integration by parts formulas that result in criteria for the existence and regularity of probability densities. Finally, it is provided an example where there are obtained conditions for the Malliavin differentiability of a particular process.