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Malliavin calculus, fall 2016

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Description 

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. The Malliavin calculus is developed similarly also on the Poisson space.

Prerequisites

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

Contents

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, Malliavin calculus in Poisson space.

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

Bibliography

 

Some other books on Malliavin Calculus

  • G. Da Prato: An introduction to Infinite-Dimensional Analysis. Springer 2006.
  • 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.
  • Bouleau N., Denis L. Dirichlet Forms Methods for Poisson Point Measures and Lévy Processes with Emphasis on the Creation-Annihilation Techniques, Springer  2015.
  • Bally V., Caramellino L., Cont R. Stochastic Integration by Parts and Functional Itô Calculus. Birkhauser 2016.
  • 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.
  • Ishikawa Y. Stochastic Calculus of Variations for Jump Processes, Springer 2016.
  • 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.
  • 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.
  • Stochastic Analysis for Poisson Point Processes Malliavin Calculus, Wiener-Itô Chaos Expansions and Stochastic Geometry. Peccati, Giovanni, Reitzner, Matthias (Editors.), Springer 2016.

Freely available lecture notes:

 

 

Teaching schedule

Weeks 36-42 and 44-50, lectures on Tuesday 12-14 in room C122 and Thursday 12-14 in room C123,

with exercise class on wednesdays 10-12 in C122. The first lecture is on tuesday 6.9 and the first

tutorials are on wednesday 14.9.

Exams:

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

Course material

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

 

Registration

 
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Exercises

Assignments

Exercise classes

GroupDayTimeRoomInstructor
1.Wednesday 10-12 C122 Dario Gasbarra 

Course feedback

Course feedback can be given at any point during the course. Click here.

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