Abstract:
The functional Ito calculus is a non-anticipative functional calculus which extends the classical Ito calculus to path-dependent functionals of a stochastic process. A key ingredient is a change of variable formula for functionals on the space of right-continuous paths (Dupire 2009, Cont & Fournié 2009, 2010), based on a notion of directional derivative proposed by B. Dupire. This formula is used to obtain explicit martingale representation formulas for a large class of martinagles. When applied to Brownian functionals, this calculus is shown to yield a non-anticipative counterpart of the Malliavin calculus.
Given an Ito process X, the functional Ito calculus allows to characterize functionals of X possessing the (local) martingale property as solutions of a new class of infinite-dimensional partial differential equations on the space of continuous paths. These functional Kolmogorov equations, which extend the well-known Kolmogorov equation to a large class of processes with path-dependent features, share many features with their finite dimensional parabolic counterparts; in particular we derive a comparison principle and uniqueness theorem for their solutions.