Stochastic Integration by Parts and Functional Itô Calculus

This volume contains the lecture notes of the two courses given by Vlad Bally and Rama Cont at the Barcelona Summer School on Stochastic Analysis in July 2012.

The notes of the course by Vlad Bally are co-authored with her collaborator Lucia Caramellino. They develop integration by parts formulas in an abstract setting, extending Malliavin's work on abstract Wiener spaces, and thereby being applicable to prove absolute continuity and regularity of the density for a broad class of random vectors.

Rama Cont's notes are on Funcional Itô Calculus, which is a non-anticipative functional calculus that extends the classical Itô calculus to path-dependent functionals of stochastic processes. As an application, a new class of path-dependent partial differential equations termed Functional Kolmogorov Equations is studied.