Functional Ito calculus and stochastic integral representation of martingales

By Rama Cont, David-Antoine Fourni'e. 2010

306 citations recorded by the source corpus.

We develop a non-anticipative calculus for functionals of a continuous semimartingale, using an extension of the Ito formula to path-dependent functionals which possess certain directional derivatives. The construction is based on a pathwise derivative, introduced by B Dupire, for functionals on the space of right-continuous functions with left limits. We show that this functional derivative admits a suitable extension to the space of square-integrable martingales. This extension denes a weak derivative which is shown to be the inverse of the Ito integral and which may be viewed as a non-anticipative \lifting" of the Malliavin derivative. These results lead to a constructive martingale representation formula for Ito processes. By contrast with the Clark-Haussmann-Ocone formula, this representation only involves nonanticipative quantities which may be computed pathwise.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/09f1ac8053c2fdb995ad9908999f9f5acf6ed5cc

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Citations in the corpus

BSDE, path-dependent PDE and nonlinear Feynman-Kac formula (2011, background): “With a very different point of view, Dupire [8, 2009] introduced a new type of a functional Itô’s formula which non-trivially generalized the classical one (See Cont and Fournié [4, 5, 6] for a more general and more systematic research).”
Path-Dependent Optimal Stochastic Control and Viscosity Solution of Associated Bellman Equations (2012, methodology): “Recently Dupire [14] introduced horizontal and vertical derivatives in the path space of a path functional, non-trivially generalized the classical Itô formula to a functional Itô formula (see Cont & Fournie [7, 8] for a more general and systematic research), and provided a functional extension of t”
Viscosity solutions to parabolic master equations and McKean–Vlasov SDEs with closed-loop controls (2018, background): “We remark that, under some stronger technical conditions, as in Cont & Fournie [13] one can show that ∂ω∂µf also does not depend on the choices of f̂ .”
Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012, background): “The main objective of this series of three papers is the following fully nonlinear parabolic path-dependent partial differential equation: { − ∂tu−G(., u, ∂ωu, ∂2ωωu) } (t, ω) = 0, (t, ω) ∈ [0, T ) × Ω. (1.1) Here Ω consists of continuous paths ω on [0, T ] starting from the origin, G is a progressi”
A martingale approach for fractional Brownian motions and related path dependent PDEs (2017, methodology): “The main idea of the proof follows that of Dupire [16] and Cont & Fournie [12].”
Note on Viscosity Solution of Path-Dependent PDE and G-Martingales (2011, unknown): “We also refer to [Cont-Fournié2010a,b] for further developments of this new calculus of Itô’s type with applications to finance.”