M A L L I A V I N C A L C U L U S - F A L L 2 0 1 5
LECTURES: Thursdays 9:30am-12:30pm, in PSY B40.
INSTRUCTOR: Solesne Bourguin. You can email me (firstname.lastname@example.org) or just drop by (MCS 226).
COURSE DESCRIPTION: The aim of the course is to introduce the important notions needed for this topic in order to make the class as self contained as possible for anyone with a strong measure theory background and solid notions of stochastic analysis. The different parts of the class could be summarized as follows:
|- Foundations of infinite dimensional Gaussian stochastic analysis: isonormal Gaussian processes, Wiener chaos, the chaos decomposition, construction and properties of multiple Wiener integrals.|
|- Malliavin calculus: definition and properties of the derivation operator and its adjoint operator (divergence operator), integration by parts, the Ornstein-Uhlenbeck semigroup and its generator, hypercontractivity property, Log-Sobolev inequalities, Meyer inequalities.|
|- Application of the theory: different versions of the integration by parts formula, quantitative limit theorems for functionals of Gaussian fields.|
|- Regularity of probability measures: Malliavin's absolute continuity criterion, Bouleau-Hirsch theorem, existence of densities and study of differentiability for solutions to stochastic differential equations and stochastic partial differential equations.|
|- Hypoellipticity and Hörmander's theorem: Lie bracket conditions, link between the regularity of SDEs and the regularity of certain classes of PDEs.|
|- Applications to finance: if time permits.|
TEXTBOOK: There is no required textbook for this class, however the following textbooks can provide a good complement to the lectures, as well as good references for additional material.
|- Nualart, D.: "The Malliavin Calculus and Related Topics", 2nd edition, Springer.|
|- Di Nunno, G., Øksendal, B., Proske, F.: "Malliavin Calculus for Lévy Processes", Springer.|
|- Peccati, G., Taqqu, M.S: "Wiener Chaos: Moments, Cumulants and Diagrams", Springer.|
GRADING: The evaluation for the class will take the form of a presentation of a research article related to the topics of the course.
HOMEWORK: Some homework will be posted on this website from time to time. These homework sets are optional and will not be taken into account in the evaluation of the class.
Problem set 1 (Chaos decomposition, multiple Wiener integrals, derivative operator, Stroock's formula).
Problem set 2 (Skohorod integral, Clark-Ocone formula, integration by parts, commutation relations).