Purely pathwise probability-free Ito integral

TL;DR

This paper introduces several simple, purely pathwise constructions of the Ito integral that do not rely on probabilistic assumptions, applicable to a wide class of functions and paths in a non-stochastic financial framework.

Contribution

It provides novel pathwise definitions of the Ito integral under various conditions, expanding the scope beyond traditional stochastic process assumptions.

Findings

Existence of the pathwise Ito integral for cadlag integrand and integrator with bounded jumps

Construction methods that do not depend on probabilistic models

Applicability in non-probabilistic financial settings

Abstract

This paper gives several simple constructions of the pathwise Ito integral $\int_0^t\phi d\omega$ for an integrand $\phi$ and a price path $\omega$ as integrator, with $\phi$ and $\omega$ satisfying various topological and analytical conditions. The definitions are purely pathwise in that neither $\phi$ nor $\omega$ are assumed to be paths of stochastic processes, and the Ito integral exists almost surely in a non-probabilistic financial sense. For example, one of the results shows the existence of $\int_0^t\phi d\omega$ for a cadlag integrand $\phi$ and a cadlag integrator $\omega$ with jumps bounded in a predictable manner.