Stochastic Integration by Parts and Functional Itô Calculus

Authors: Vlad Bally, Lucia Caramellino, Rama Cont
Editors: Frederic Utzet, Josep Vives



Includes a general method for proving existence of a density for stochastic processes, using interpolation spaces

Illustrates a pathwise derivation of the Ito formula and the Functional Ito calculus

Provides solutions to problems in applied fields such as mathematical finance

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.

Table of contents

Integration by parts formulas and the Riesz transform

Construction of integration by parts formulas

Regularity of probability laws by using an interpolation method

Overview

Pathwise calculus for non-anticipative functionals

The functional Itô formula

Weak functional calculus for square-integrable processes

Functional Kolmogorov equations