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Wed, 28.09.2011, 16:15, seminar room C-209, UZA 4 University of Vienna, Nordbergstrasse 15, 1040, Wien

Rama Cont (Columbia University) "Functional Ito Calculus"

Abstract: We develop a non-anticipative functional calculus which extends the Ito calculus to path-dependent functionals of right- continuous semimartingales [1, 3], using a notion of non-anticipative functional derivative introduced by B. Dupire [6]. This calculus is shown to be, in a precise sense, a non-anticipative analogue of the Malliavin calculus; however, our construction holds for a large class of semimartingales and makes no use of the Gaussian properties of the Wiener space.

Our framework, which is sufficiently general to cover functionals depending on quadratic variation and involving exit times of a process, is used to obtain several new results. First, we obtain a martingale representation formula for square integrable functionals of a semimartingale [2]. Second, we characterize local martingales which satisfy a regularity property as solutions of a functional differential equation, for which existence and uniqueness results are given [5]. These results have natural applications in stochastic control and mathematical finance: they allow to derive a universal pricing equation and a general hedging formula for path-dependent options, and reformulate Backward Stochastic Differential Equations (BSDEs) as PDEs on path space.

Based on joint work with David Fournie (Columbia University).

References [1] R Cont and D Fournie (2010) A functional extension of the Ito formula, Comptes Rendus de l'Academie des Sciences, Volume 348, Issues 1-2, January 2010, Pages 57-61. [2] R Cont and D Fournie (2009) Functional Ito calculus and stochastic integral representation of martingales, http: //arxiv.org/abs/1002.2446. To appear in: Annals of Probability. [3] R Cont and Fournie (2010) Change of variable formulas for non-anticipative functionals on path space, Journal of Functional Analysis, Volume 259, No 4, Pages 1043-1072. [4] R Cont (2010) Numerical computation of martingale representations, Working Paper. [5] B Dupire (2009) Functional Ito calculus, www.ssrn.com. [6] R Cont, D Fournie (2010) Martingales and functional differential equations, Working Paper.

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