Stochastic Integration by Parts and Functional Itô Calculus / by Vlad Bally, Lucia Caramellino, Rama Cont ; edited by Frederic Utzet, Josep Vives.

Format:

Book

Author/Creator:

Bally, Vlad., Author.

Caramellino, Lucia., Author.

Cont, Rama., Author.

Contributor:

Utzet, Frederic., Editor.

Vives, Josep., Editor.

Series:

Advanced Courses in Mathematics - CRM Barcelona, 2297-0304

Language:

English

Subjects (All):

Probabilities.

Differential equations.

Differential equations, Partial.

Probability Theory and Stochastic Processes.

Ordinary Differential Equations.

Partial Differential Equations.

Local Subjects:

Probability Theory and Stochastic Processes.

Ordinary Differential Equations.

Partial Differential Equations.

Physical Description:

1 online resource (IX, 207 p. 1 illus. in color.)

Edition:

1st ed. 2016.

Place of Publication:

Cham : Springer International Publishing : Imprint: Birkhäuser, 2016.

Language Note:

English

Summary:

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.

Contents:

Intro

Foreword

Contents

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

Preface

Problem 1

Problem 2

Problem 3

Problem 4

Conclusion

Chapter 1 Integration by parts formulas and the Riesz transform

1.1 Sobolev spaces associated to probability measures

1.2 The Riesz transform

1.3 A first absolute continuity criterion: Malliavin-Thalmaier representation formula

1.4 Estimate of the Riesz transform

1.5 Regularity of the density

1.6 Estimate of the tails of the density

1.7 Local integration by parts formulas and local densities

1.8 Random variables

Chapter 2 Construction of integration by parts formulas

2.1 Construction of integration by parts formulas

2.1.1 Derivative operators

2.1.2 Duality and integration by parts formulas

2.1.3 Estimation of the weights

Iterated derivative operators, Sobolev norms

Estimate of |γ(F)|l

Bounds for the weights Hqβ (F,G)

2.1.4 Norms and weights

2.2 Short introduction to Malliavin calculus

2.2.1 Differential operators

Step 1: Finite-dimensional di erential calculus in dimension n

Step 2: Finite-dimensional di erential calculus in arbitrary dimension

Step 3: Infinite-dimensional calculus

2.2.2 Computation rules and integration by parts formulas

2.3 Representation and estimates for the density

2.4 Comparisons between density functions

2.4.1 Localized representation formulas for the density

2.4.2 The distance between density functions

2.5 Convergence in total variation for a sequence of Wiener functionals

Chapter 3 Regularity of probability laws by using an interpolation method

3.1 Notations

3.2 Criterion for the regularity of a probability law

3.3 Random variables and integration by parts

3.4 Examples

3.4.1 Path dependent SDE's.

3.4.2 Diffusion processes

3.4.3 Stochastic heat equation

3.5 Appendix A: Hermite expansions and density estimates

3.6 Appendix B: Interpolation spaces

3.7 Appendix C: Superkernels

Bibliography

Part II Functional Itô Calculus and Functional Kolmogorov Equations

Chapter 4 Overview

4.1 Functional Itô Calculus

4.2 Martingale representation formulas

4.3 Functional Kolmogorov equations and path dependent PDEs

4.4 Outline

Notations

Chapter 5 Pathwise calculus for non-anticipative functionals

5.1 Non-anticipative functionals

5.2 Horizontal and vertical derivatives

5.2.1 Horizontal derivative

5.2.2 Vertical derivative

5.2.3 Regular functionals

5.3 Pathwise integration and functional change of variable formula

5.3.1 Quadratic variation of a path along a sequence of partitions

5.3.2 Functional change of variable formula

5.3.3 Pathwise integration for paths of finite quadratic variation

5.4 Functionals defined on continuous paths

5.5 Application to functionals of stochastic processes

Chapter 6 The functional Itô formula

6.1 Semimartingales and quadratic variation

6.2 The functional Itô formula

6.3 Functionals with dependence on quadratic variation

Chapter 7 Weak functional calculus for square-integrable processes

7.1 Vertical derivative of an adapted process

7.2 Martingale representation formula

7.3 Weak derivative for square integrable functionals

7.4 Relation with the Malliavin derivative

7.5 Extension to semimartingales

7.6 Changing the reference martingale

7.7 Forward-Backward SDEs

Chapter 8 Functional Kolmogorov equations

8.1 Functional Kolmogorov equations and harmonic functionals

8.1.1 Stochastic differential equations with path dependent coefficients

8.1.2 Local martingales and harmonic functionals.

8.1.3 Sub-solutions and super-solutions

8.1.4 Comparison principle and uniqueness

8.1.5 Feynman-Kac formula for path dependent functionals

8.2 FBSDEs and semilinear functional PDEs

8.3 Non-Markovian stochastic control and path dependent HJB equations

8.4 Weak solutions

Comments and references

Bibliography.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Description based on publisher supplied metadata and other sources.

ISBN:

3-319-27128-8

OCLC:

1066194701