Functional Itô calculus†

By Bruno Dupire. Quantitative finance (Print), 2019

295 citations recorded by the source corpus.

We extend some results of the Itô calculus to functionals of the current path of a process to reflect the fact that often the impact of randomness is cumulative and depends on the history of the process, not merely on its current value. We express the differential of the functional in terms of adequately defined partial derivatives to obtain an Itô formula. We develop an extension of the Feynman-Kac formula to the functional case and an explicit expression of the integrand in the Martingale Representation Theorem. We establish that under certain conditions, even path dependent options prices satisfy a partial differential equation in a local sense. We exploit this fact to find an expression of the price difference between two models and compute variational derivatives with respect to the volatility surface.

Subfields: Finance

View Paper: https://www.semanticscholar.org/paper/4b61c46dfb0d02a204a916fe1899591081bb2d86

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Citations in the corpus

Optimal control of path-dependent McKean–Vlasov SDEs in infinite-dimension (2020, methodology): “We then define a notion of pathwise measure derivative, which extends the Wasserstein derivative due to Lions [62], and prove a related functional Itô formula in the spirit of Dupire [33] and Wu and Zhang [73].”
Viscosity solutions to parabolic master equations and McKean–Vlasov SDEs with closed-loop controls (2018, methodology): “The main result in this regard is a functional Itô formula in the McKean-Vlasov setting, extending the well known result of Dupire [15] in the standard setting.”
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”
A martingale approach for fractional Brownian motions and related path dependent PDEs (2017, methodology): “In particular, unlike in [16] where one needs only to consider the stopped paths, here we need to concatenate the observed path up to the current time with a certain smooth observable curve derived from the distribution of the future paths.”
Wellposedness of Second Order Master Equations for Mean Field Games with Nonsmooth Data (2019, unknown): “Here the variable (t, x, ω) ∈ [0, T ] × R × Ω0, and ∂ωu is the path derivative introduced by [33].”
Note on Viscosity Solution of Path-Dependent PDE and G-Martingales (2011, methodology): “The following notations are mainly from [Dupire2009].”
Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity (2018, methodology): “3 Extension to path-dependent functionals An important generalization of Föllmer’s pathwise Itô formula is to the case of path-dependent functionals [8] of paths S ∈ V2(π) using Dupire’s notion of functional derivative [12]; see [7] for an overview.”