On the Functional L\'{e}vy-It\^{o} Stochastic Calculus

TL;DR

This paper extends the functional Itô's calculus by defining derivatives along functionals and develops an optimal local-time based Itô's formula for Lévy processes, with some applications.

Contribution

It introduces a new notion of derivative along functionals and derives an optimal local-time based Itô's formula for Lévy processes.

Findings

Extended functional Itô's formula with derivative along functionals

Derived an optimal local-time based Itô's formula for Lévy processes

Provided applications demonstrating the new 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.