## Functional Ito Calculus and Applications - Lecture 2: Weak functional calculus and martingale representation

#### Rama ContImperial College

OUTLINE: The Functional Ito Calculus [1, 2, 3, 8] is a non-anticipative calculus for functionals defined on spaces of paths; it extends many results of Ito's stochastic calculus to path-dependent functionals of semimartingales [1, 2, 3]. These lectures provide an introduction to this theory, its links with Malliavin calculus, and a glimpse of applications in mathematical finance and stochastic control theory. The starting point of the theory is a functional extension of the Ito formula [1, 8], which has numerous applications in stochastic analysis and mathematical finance. Two ingredients of the proof are Föllmer's deterministic proof of the Ito formula  and a notion of pathwise functional derivative proposed by B. Dupire . The functional Ito formula may be used to obtain explicit martingale representation formulas [2, 5] for functionals of a square-integrable martingale S. By contrast with the Clark-Ocone formula, which is based on Malliavin calculus, these formulas are based on non-anticipative quantities which may be computed pathwise using simple numerical schemes . In mathematical finance, these formulas allow explicit computations of hedging strategies for path-dependent derivatives [6, 7]. The martingale representation formula  is the starting point for the construction of a weak functional calculus [2, 5], applicable to all square-integrable semimartingales. This weak functional calculus allows to define the functional derivative of a process adapted to a filtration with respect to a reference process generating the filtration. The functional Ito calculus gives rise to a new class of 'path-dependent' partial differential equations for functionals on path space, which share many properties with parabolic PDEs on finite dimensional spaces. We show that a large class of martingales may be characterized as solutions to functional Kolmogorov equations, which lead to the notion of harmonic functional on path space. We study existence, uniqueness and comparison properties of solutions for such equations and show that they lead to Feynman-Kac formulas for path-dependent functionals of a square-integrable Brownian martingales. These functional PDEs exhibit a natural link with Forward-Backward systems of stochastic differential equations. These results have various applications to non-Markovian stochastic control, forward-backward stochastic differential equations and the pricing and hedging of path-dependent contingent claims [5, 6, 7].

TALKS: Lecture 1) Pathwise calculus for non-anticipative functionals; Lecture 2) Weak functional calculus and martingale representation formulas; Lecture 3) Functional Kolmogorov equations and harmonic functionals on path space; Lecture 4) Application to Forward-Backward stochastic differential equations

REFERENCES:  R Cont and D Fournié (2010) A functional extension of the Ito formula, Comptes Rendus de l'Académie des Sciences, Volume 348, Issues 1-2, January 2010, Pages 57-61.  R Cont and D Fournié (2009) Functional Ito calculus and stochastic integral representation of martingales, to appear in Annals of probability, http://arxiv.org/abs/1002.2446.  R Cont and D Fournié (2010) Change of variable formulas for non-anticipative functionals on path space, Journal of Functional Analysis, Volume 259, No 4, Pages 1043-1072.  R Cont and D Fournié (2010) Functional Kolmogorov equations, Working Paper.  R Cont (2012) Functional Ito Calculus and Functional Kolmogorov equations, Lecture Notes of the Barcelona Summer School on Stochastic Analysis (July 2012).  R Cont and C Riga (2014) Pathwise analysis and robustness of hedging strategies for path-dependent derivatives, Working Paper.  R Cont and Y Lu (2014) Weak approximation of martingale representations, http://arxiv.
org/abs/1501.00383.  B Dupire (2009) Functional Ito calculus, www.ssrn.com.  H. Föllmer (1981) Calcul d'Itô sans probabilités, Séminaire de Probabilités Vol. XV, Springer, Berlin, pp. 143150.

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