Malliavin Calculus, fall 2013
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.
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:
- Friz, Peter: An introduction to Malliavin calculus.
- Øksendal, Bernt: An introduction to Malliavin calculus with applications to economics.
- Sottinen Tommi Malliavin-laskenta
- Imkeller Peter Malliavin's calculus and applications in stochastic control and finance, Warsaw 2008
- Üstünel, Ali Süleyman Analysis on Wiener Space and Applications (Arxiv:1003.1649, 2010).
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