© 2016

Stochastic Integration by Parts and Functional Itô Calculus

  • Frederic Utzet
  • Josep Vives
  • Includes a general method for

  • proving existence of a density for stochastic processes, using interpolation

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

  • Provides solutions to problems in applied fields such as mathematical finance


Part of the Advanced Courses in Mathematics - CRM Barcelona book series (ACMBIRK)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Integration by Parts Formulas, Malliavin Calculus, and Regularity of Probability Laws

    1. Front Matter
      Pages 1-7
    2. Vlad Bally, Lucia Caramellino
      Pages 9-31
    3. Vlad Bally, Lucia Caramellino
      Pages 33-81
    4. Vlad Bally, Lucia Caramellino
      Pages 83-114
  3. Functional Itô Calculus and Functional Kolmogorov Equations

    1. Front Matter
      Pages 115-117
    2. Rama Cont
      Pages 119-123
    3. Rama Cont
      Pages 153-162
    4. Rama Cont
      Pages 183-207
  4. Back Matter
    Pages 208-208

About this book


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.


Malliavin calculus probability laws path-dependent PDE Kolmogorov equations interpolation spaces ordinary differential equations

Authors and affiliations

  1. 1.Cité Descartes 5Université de Marne-la-ValléeMarne la ValléeFrance
  2. 2.Dipartimento di MatematicaUniversità di Roma Tor VergataRomaItaly
  3. 3.Department of MathematicsImperial CollegeLondonUK

Editors and affiliations

  • Frederic Utzet
    • 1
  • Josep Vives
    • 2
  1. 1.Departament de MatemàtiquesUniversitat Autònoma de Barcelona Departament de MatemàtiquesBellaterraSpain
  2. 2.Faculty of MathematicsUniversity of Barcelona Faculty of MathematicsBarcelonaSpain

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