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

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

Rama Cont's notes provide an
introduction to the Functional Itô Calculus, a non-anticipative functional
calculus that extends the classical Itô calculus to path-dependent functionals
of stochastic processes. This calculus leads to a new class of path-dependent
partial differential equations, termed Functional Kolmogorov Equations, which
arise in the study of martingales and forward-backward stochastic differential
equations.

This book will appeal to both young and senior researchers in probability and stochastic processes, as well as to practitioners in mathematical finance.