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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Integration by Parts Formulas, Malliavin Calculus, and Regularity of Probability Laws

Frontmatter

Chapter 1. Integration by parts formulas and the Riesz transform

Abstract
The aim of this chapter is to develop a general theory allowing to study the existence and regularity of the density of a probability law starting from integration by parts type formulas (leading to general Sobolev spaces) and the Riesz transform, as done in [2].
Vlad Bally, Lucia Caramellino

Chapter 2. Construction of integration by parts formulas

Abstract
In this chapter we construct integration by parts formulas in an abstract framework based on a finite-dimensional random vector \( V = \left( {V_1 , \ldots ,V_J } \right); \) we follow [8]. Such formulas have been used in [8, 10] in order to study the regularity of solutions of jump-type stochastic equations, that is, including equations with discontinuous coefficients for which the Malliavin calculus developed by Bismut [17] and by Bichteler, Gravereaux, and Jacod [16] fails.
Vlad Bally, Lucia Caramellino

Chapter 3. Regularity of probability laws by using an interpolation method

Abstract
One of the outstanding applications of Malliavin calculus is the criterion of regularity of the law of a functional on the Wiener space (presented in Section 2.3). The functional involved in such a criterion has to be regular in Malliavin sense, i.e., it has to belong to the domain of the differential operators in this calculus. As long as solutions of stochastic equations are concerned, this amounts to regularity properties of the coefficients of the equation.
Vlad Bally, Lucia Caramellino

Functional Itô Calculus and Functional Kolmogorov Equations

Frontmatter

Chapter 4. Overview

Abstract
Many questions in stochastic analysis and its applications in statistics of processes, physics, or mathematical finance involve path-dependent functionals of stochastic processes and there has been a sustained interest in developing an analytical framework for the systematic study of such path-dependent functionals.
Rama Cont

Chapter 5. Pathwise calculus for non-anticipative functionals

Abstract
The focus of these lectures is to define a calculus which can be used to describe the variations of interesting classes of functionals of a given reference stochastic process X. In order to cover interesting examples of processes, we allow X to have right-continuous paths with left limits, i.e., its paths lie in the space \( D\left( {\left[ {0,\,T} \right],\,{\mathbb{R}}^d } \right) \) of càdlàg paths.
Rama Cont

Chapter 6. The functional Itô formula

Without Abstract
Rama Cont

Chapter 7. Weak functional calculus for square-integrable processes

Abstract
The pathwise functional calculus presented in Chapters 5 and 6 extends the Itô Calculus to a large class of path dependent functionals of semimartingales, of which we have already given several examples.
Rama Cont

Chapter 8. Functional Kolmogorov equations

Abstract
One of the key topics in Stochastic Analysis is the deep link between Markov processes and partial differential equations, which can be used to characterize a diffusion process in terms of its infinitesimal generator [69].
Rama Cont

Backmatter

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