Causal functional calculus

TL;DR

This paper develops a new topology and calculus for causal functionals on path spaces, enabling pathwise integration and change of variable formulas that extend classical results to broader classes of functionals, including those with finite quadratic variation.

Contribution

It introduces a novel topology and a pathwise calculus for causal functionals, extending existing stochastic calculus results to more general path spaces and functionals.

Findings

Extended Föllmer-Ito calculus to paths with finite quadratic variation.

Developed a pathwise change of variable formula for a broad class of functionals.

Represented harmonic functionals as pathwise integrals of closed 1-forms.

Abstract

We construct a new topology on the space of stopped paths and introduce a calculus for causal functionals on generic domains of this space. We propose a generic approach to pathwise integration without any assumption on the variation index of a path and obtain functional change of variable formulas which extend the results of \follmer\ (1981) and Cont \& Fourni\'e (2010) to a larger class of functionals, including \follmer's pathwise integrals. We show that a class of smooth functionals possess a pathwise analogue of the martingale property. For paths that possess finite quadratic variation, our approach extends F\"ollmer-Ito calculus and removes previous restriction on the time partition sequence. We introduce a foliation structure on this path space and show that harmonic functionals may be represented as pathwise integrals of closed 1-forms.