Wiki source code of Malliavin Calculus fall 2013

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1 = Malliavin Calculus, fall 2013 =
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3 === Lecturer ===
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5 === {{mention reference="XWiki.gasbarra@helsinki_fi" style="FULL_NAME" anchor="XWiki-gasbarra@helsinki_fi-qoETD"/}} ===
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7 ==== (% style="color: rgb(51,51,51);font-size: 14.0pt;font-weight: bold;line-height: normal;" %)Scope(%%) ====
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9 10 sp.
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11 === Type ===
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13 Advanced studies
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15 === Prerequisites ===
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17 Probability theory or Measure and Integration. The background in functional and stochastic analysis will be presented in the lectures
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19 === Lectures ===
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21 During weeks 36-42 and 44-50 on mondays at 10-12 in C123 and fridays at 10-12 in B321. (% style="font-size: 10.0pt;line-height: 13.0pt;" %)The first lecture will be on friday 6.9.
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23 === Exams ===
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25 The exam is passed by solving the problems assigned weekly and writing a final home exam:
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27 === [[Home exam paper (.pdf)>>attach:home_exam_malliavin_2014.pdf]] [[Home exam pap>>attach:home_exam_malliavin_2014.tex]][[er (.tex)>>attach:home_exam_malliavin_2014.tex]](% style="font-size: 10.0pt;line-height: 13.0pt;" %) (%%) ===
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29 === Description ===
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31 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 (% style="font-size: 10.0pt;line-height: 13.0pt;" %)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.
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33 === Contents ===
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35 Introduction: Malliavin calculus in finite dimension. Gaussian random vectors. Wick's Gaussian moment formula.Malliavin-Sobolev space, (% style="font-size: 10.0pt;line-height: 13.0pt;" %)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.
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37 (% style="font-size: 10.0pt;line-height: 13.0pt;" %) 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.
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39 Applications: computation and smoothness of densities of random variables and solutions of stochastics differential equations.(% style="font-size: 10.0pt;line-height: 13.0pt;" %) Option pricing and computation of sensitivities.
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41 === Lecture materials: ===
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43 [[slides: ABC of malliavin calculus >>attach:malliavin_slides_2013.pdf]] .Lecture notes: [[Notes on Gaussian measures in infinite dimension>>attach:complements_banach_2013.pdf]]
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46
47 === Bibliography ===
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49 * Nualart, David: //[[The Malliavin calculus and related topics>>url:http://www.amazon.com/gp/product/038794432X/sr=1-2/qid=1137166995/ref=pd_bbs_2/104-6259552-5751953?%5Fencoding=UTF8||shape="rect"]], 2nd Edition. //Probability and its Applications, Springer-Verlag Berlin-Heidelberg, 2006.
50 * G. Da Prato: An introduction to Infinite-Dimensional Analysis. Springer 2006.
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52 (% style="text-align: left;" %)
53 Some other books on Malliavin Calculus
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55 (% style="text-align: left;" %)
56 * (% style="font-size: 10.0pt;line-height: 13.0pt;" %)G. Da Prato: Introducion to Stochastic analysis and Malliavin calculus, Edizioni della Normale, Pisa 2007.
57 * (% style="font-size: 10.0pt;line-height: 13.0pt;" %)D. Bell : The Malliavin calculus. Pitman Monographs and Surveys in Pure and Applied Mathematics 34, 1987.
58 * (% style="font-size: 10.0pt;line-height: 13.0pt;" %)Bouleau N. //Error Calculus for Finance and Physics: The Language of Dirichlet Forms// , De Gruyter Expositions in Mathematics, 2003.
59 * (% style="font-size: 10.0pt;line-height: 13.0pt;" %)Bouleau N., Hirsch F. //Dirichlet Forms and Analysis on Wiener Space// , De Gruyter Studies in Mathematics, 1991.
60 * (% style="font-size: 10.0pt;line-height: 13.0pt;" %)R. Carmona, M. Tehranchi: Interest Rate Models, An Infinite-dimensional Stochastic Analysis Perspective, 2006.
61 * (% style="font-size: 10.0pt;line-height: 13.0pt;" %)G. Di Nunno, B. Øksendal, F. Proske: Malliavin Calculus for Lévy Processes with Applications to Finance, 2009.
62 * (% style="font-size: 10.0pt;line-height: 13.0pt;" %)Malliavin P. Thalmaier A.: //Stochastic Calculus of Variations in Mathematical Finance// , Springer Finance, 2006.
63 * (% style="font-size: 10.0pt;line-height: 13.0pt;" %)Peccati G, Taqqu M:[[ Wiener Chaos,Moments,Cumulants and Diagrams, a Survey with Computer Implementation.>>url:http://link.springer.com/book/10.1007/978-88-470-1679-8/page/1||shape="rect"]] Springer & Bocconi series 1, 2011.
64 * (% style="font-size: 10.0pt;line-height: 13.0pt;" %)Nourdin Ivan: Selected Aspects of Fractional Brownian Motion, Bocconi & Springer series 4, 2012.
65 * (% style="font-size: 10.0pt;line-height: 13.0pt;" %)Nourdin Ivan, Peccati Giovanni: Normal Approximations with Malliavin Calculus: From Stein's Method to Universality. Cambridge Tracts in Mathematics, 2012.
66 * (% style="font-size: 10.0pt;line-height: 13.0pt;" %)Privault N. Stochastic Analysis in Discrete and Continuous Settings, with Normal Martingales, Springer 2009.
67 * (% style="font-size: 10.0pt;line-height: 13.0pt;" %)Shigekawa I., //Stochastic analysis// , AMS 2004.
68 * (% style="font-size: 10.0pt;line-height: 13.0pt;" %)Malliavin, Paul: [[(% style="font-size: 10.0pt;line-height: 13.0pt;" %)//Stochastic analysis//>>url:http://www.amazon.com/gp/product/3540570241/sr=1-12/qid=1137166995/ref=sr_1_12/104-6259552-5751953?%5Fencoding=UTF8||shape="rect"]](%%)//.// Grundlehren der Mathematischen Wissenschaften, 313. Springer-Verlag, Berlin 1997.
69 * (% style="font-size: 10.0pt;line-height: 13.0pt;" %)Malliavin Paul, L Kay, H Airault,G Letac. Integration and Probability. Springer Graduate Text in Mathematics, 1995.
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71 (% style="text-align: left;" %)
72 Freely available lecture notes:
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74 * Friz, Peter: [[//An introduction to Malliavin calculus//>>url:http://www.statslab.cam.ac.uk/~~peter/malliavin/Malliavin2005/mall.pdf||shape="rect"]]//.//
75 * Øksendal, Bernt: [[//An introduction to Malliavin calculus with applications to economics//>>url:http://www.quantcode.com/modules/wflinks/visit.php?cid=11&lid=4||shape="rect"]]//.//
76 * //Sottinen Tommi [[Malliavin-laskenta>>url:http://lipas.uwasa.fi/~~tsottine/lecture_notes/malliavin.pdf||shape="rect"]] //
77 * //Imkeller Peter [[Malliavin's calculus and applications in stochastic control and finance, Warsaw 2008>>url:http://wws.mathematik.hu-berlin.de/~~imkeller/teaching/lectures/Supplement/warschau_malliavin2.pdf||shape="rect"]]
78 //
79 * Üstünel, Ali Süleyman  [[Analysis on Wiener Space and Applications>>url:http://arxiv.org/abs/1003.1649||shape="rect"]] (Arxiv:1003.1649, 2010).
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