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Abstract: A non-anticipative functional on a function space D is a family (F _{t})_{t≥0} of functionals which represents the time evolution of quantity which depends in a causal manner on the path (omega ∈ D) of an observable. We develop a differential calculus for such non-anticipative functionals [3], using a notion of functional derivative proposed by B Dupire. Our framework only requires the existence of certain directional derivatives and covers functionals deﬁned on spaces of irregular paths with inﬁnite variation. Applied to stochastic processes, this functional calculus extends the Ito calculus to path-dependent functionals of right- continuous semimartingales [1, 2]. In the case of Brownian motion, it yields a non-anticipative analogue of the Malliavin calculus [2]; however, the construction makes no use of the Gaussian properties of the Wiener space and holds for a large class of processes. Several probabilistic results are derived using this framework. First, we obtain a martingale representation formula for square integrable functionals of an Ito process [2]. Second, we characterize martingales which satisfy a regularity property as solutions of a (deterministic) functional differential equation, for which a uniqueness result is given [4]. These results have natural applications in stochastic control and mathematical ﬁnance, which we will brieﬂy sketch.
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, to appear in Annals of probability, http://arxiv.org/abs/1002.2446.
[3] R Cont and D 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 and D Fournie (2010) Functional Kolmogorov equations and harmonic functionals, Working Paper.
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Donnerstag, den 12. Mai 2011 um 17:15 Uhr, in INF 288, HS 2 Donnerstag, den 12. Mai 2011 at 17:15, in INF 288, HS 2