On the Functional Lévy-Itô Stochastic Calculus

Abstract: Several versions of It^{o}'s formula have been obtained in the context of the functional stochastic calculus. Here, we revisit this topic in two ways. First, by defining a notion of derivative along a functional, we extend the setting of the (semimartingale) functional It^{o}'s formula and corresponding calculus. Second, for L\'{e}vy processes, an optimal local-time based It^{o}'s formula is obtained. Some quick applications are then given.