Remarks on F\"ollmer's pathwise It\^o calculus

TL;DR

This paper extends Föllmer's pathwise Itô calculus to càdlàg paths with quadratic variation, exploring fundamental properties, integral equations, and their solutions within this generalized framework.

Contribution

It generalizes Föllmer's pathwise Itô calculus from continuous to càdlàg paths with quadratic variation, including properties and solvability of integral equations.

Findings

Established associativity and integration by parts for càdlàg paths

Demonstrated solvability of certain integral equations within the pathwise calculus

Extended fundamental results of Itô calculus to a broader class of paths

Abstract

We extend some results about F\"ollmer's pathwise It\^o calculus that have only been derived for continuous paths to c\`adl\`ag paths with quadratic variation. We study some fundamental properties of pathwise It\^o integrals with respect to c\`adl\`ag integrators, especially associativity and the integration by parts formula. Moreover, we study integral equations with respect to pathwise It\^o integrals. We prove that some classes of integral equations, which can be explicitly solved in the usual stochastic calculus, can also be solved within the framework of F\"ollmer's calculus.