Pathwise integration with respect to paths of finite quadratic variation

TL;DR

This paper develops a pathwise integral for paths with finite quadratic variation, establishing an isometry property and a decomposition into signal and noise components, extending stochastic calculus concepts to a deterministic setting.

Contribution

It introduces a new pathwise integral with an isometry property and a signal-plus-noise decomposition for irregular paths, advancing deterministic analysis of quadratic variation.

Findings

Established a pathwise isometry property analogous to Ito's isometry.

Represented the integral as a continuous map on a vector space of integrands.

Derived a unique 'signal plus noise' decomposition for functionals of irregular paths.

Abstract

We study a pathwise integral with respect to paths of finite quadratic variation, defined as the limit of non-anticipative Riemann sums for gradient-type integrands. We show that the integral satisfies a pathwise isometry property, analogous to the well-known Ito isometry for stochastic integrals. This property is then used to represent the integral as a continuous map on an appropriately defined vector space of integrands. Finally, we obtain a pathwise 'signal plus noise' decomposition for regular functionals of an irregular path with non-vanishing quadratic variation, as a unique sum of a pathwise integral and a component with zero quadratic variation.