A functional extension of the Ito formula

By Rama Cont, David-Antoine Fournié. 2010

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Optimal stopping with signatures (2021, methodology): “Following [KLPA20], we will use a distance motivated by Dupire’s functional It¯o calculus, see [Dup19, CF10].”
BSDE, path-dependent PDE and nonlinear Feynman-Kac formula (2011, methodology): “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).”
Change of variable formulas for non-anticipative functionals on path space ✩ (2010, unknown): “Pathwise derivatives of non-anticipative functionals We define two notions of pathwise derivatives for a non-anticipative functional F = (Ft )t∈[0,T ]: the horizontal derivative, which is a derivative with respect to time, and the vertical derivative, which is a derivative with respect to the curren”
Functional Ito calculus and stochastic integral representation of martingales (2010, background): “Building on this insight, we develop hereafter a nonanticipative calculus [5] for a class of processes—including the above examples—which may be represented as Y(t) = Ft ({X(u),0 ≤ u ≤ t}, {A(u),0 ≤ u ≤ t}) = Ft(Xt ,At), (2) where A is the local quadratic variation defined by [X](t) = ∫ t 0 A(u)du, ”