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An elementary introduction to Malliavin calculus

Vlad Bally 1
1 MATHFI - Financial mathematics
Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech, UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12
Abstract : We give an introduction to Malliavin calculus following the notes of four lectures that I gave in the working group of the research team Mathfi in October 2001. These notes contain three topics: An elementary presentation of Malliavin's differential operators and of the integration by parts formula which represents the central tool in this theory. The Wiener chaos decomposition and the differential operators of Malliavin calculus as operators on the Wiener space. The application of this calculus to the study of the density of the law of a diffusion process. This was the initial application of Malliavin's calculus - and provides a probabili- stic proof of Hormander's hypothesis. The aim of these notes is to emphasis some basic ideas and technics related to this theory and not to provide a text book. The book of Nualart, for example, is an excellent nomography on this topic. So we live out some technical points (we send to the papers or books where complete proofs may be found) and do not treat problems in all generality or under the more general assumptions. We hope that this choice provides a soft presentation of the subject which permits to understand better the objects and the ideas coming on as well as the possible applications.
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https://hal.inria.fr/inria-00071868
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Submitted on : Tuesday, May 23, 2006 - 7:03:35 PM
Last modification on : Wednesday, September 4, 2019 - 1:52:07 PM
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Vlad Bally. An elementary introduction to Malliavin calculus. [Research Report] RR-4718, INRIA. 2003. ⟨inria-00071868⟩

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