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Malliavin Calculus, fall 2013


Dario Gasbarra


10 sp.


Advanced studies


Probability theory or Measure and Integration. The background in functional and stochastic analysis will be presented in the lectures


During weeks 36-42 and 44-50 on mondays at 10-12 in C123 and fridays at 10-12 in  B321. The first lecture will be on friday 6.9.


The exam is passed by solving the  problems assigned weekly and writing a final home exam:

Home exam paper (.pdf)  Home exam paper (.tex)  


In Stochastic analysis  a central role is played by the stochastic integral with respect to Brownian motion, introduced by Ito (1944). The classical Theory of Frechet derivatives on a Banach space does not fit with Ito integration. In 1976 Paul Malliavin invented a new way to differentiate functionals  of Brownian motion. The adjoint operator of the Malliavin derivative is the Skorokhod integral, which extends the Ito integral to non-anticipative integrands. Malliavin calculus has concrete applications, for example in mathematical finance: the Ito-Clarck-Ocone formula gives explicitely the martingale representation of a square integrable Brownian functional.This is used to compute the hedging of path-dependent options.  


Introduction: Malliavin calculus in finite dimension. Gaussian random vectors. Wick's Gaussian moment formula.Malliavin-Sobolev space, Gaussian integration by parts and Hermite polynomials. Gaussian measures on Banach and Hilbert spaces. Reproducing Kernel Hilbert Space, Wiener integral, Isonormal Gaussian process. Example: Brownian motion.

 Iton isometry and Ito integral, Girsanov's theorem. Multiple Wiener integrals and Chaos expansion. Malliavin derivative  and Skorokhod's divergence integral. Ito-Clarck-Ocone representation. Pathwise non-anticipative integrals. Malliavin calculus for fractional Brownian motion.

Applications: computation and smoothness of densities of random variables and solutions of stochastics differential equations. Option pricing and  computation of sensitivities. 

Lecture materials:

slides: ABC of malliavin calculus  .Lecture notes: Notes on Gaussian measures in infinite dimension



  • Nualart, David: The Malliavin calculus and related topics, 2nd Edition. Probability and its Applications, Springer-Verlag Berlin-Heidelberg, 2006.
  • G. Da Prato: An introduction to Infinite-Dimensional Analysis. Springer 2006.

Some other books on Malliavin Calculus

  • G. Da Prato: Introducion to Stochastic analysis and Malliavin calculus, Edizioni della Normale, Pisa 2007.
  • D. Bell : The Malliavin calculus. Pitman Monographs and Surveys in Pure and Applied Mathematics 34, 1987.
  • Bouleau N. Error Calculus for Finance and Physics: The Language of Dirichlet Forms , De Gruyter Expositions in Mathematics, 2003.
  • Bouleau N., Hirsch F. Dirichlet Forms and Analysis on Wiener Space , De Gruyter Studies in Mathematics, 1991.
  • R. Carmona, M. Tehranchi: Interest Rate Models, An Infinite-dimensional Stochastic Analysis Perspective, 2006.
  • G. Di Nunno, B. Øksendal, F. Proske: Malliavin Calculus for Lévy Processes with Applications to Finance, 2009.
  • Malliavin P. Thalmaier A.: Stochastic Calculus of Variations in Mathematical Finance , Springer Finance, 2006.
  • Peccati G, Taqqu M: Wiener Chaos,Moments,Cumulants and Diagrams, a Survey with Computer Implementation. Springer & Bocconi series 1, 2011.
  • Nourdin Ivan: Selected Aspects of Fractional Brownian Motion,  Bocconi & Springer series 4, 2012.
  • Nourdin Ivan, Peccati Giovanni: Normal Approximations with Malliavin Calculus: From Stein's Method to Universality. Cambridge Tracts in Mathematics, 2012.
  • Privault N. Stochastic Analysis in Discrete and Continuous Settings, with Normal Martingales, Springer 2009.
  • Shigekawa I., Stochastic analysis , AMS 2004.
  • Malliavin, Paul: Stochastic analysis. Grundlehren der Mathematischen Wissenschaften, 313. Springer-Verlag, Berlin 1997.
  • Malliavin Paul, L Kay, H Airault,G Letac.  Integration and Probability.  Springer Graduate Text in Mathematics, 1995.

Freely available lecture notes:



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