A rough path approach to pathwise stochastic integration \`a la F\"ollmer

TL;DR

This paper introduces a comprehensive framework for pathwise stochastic integration using rough path theory, extending classical methods to more general integrands and establishing conditions for invariance of quadratic variation and L9vy area.

Contribution

It develops a general pathwise stochastic integration framework that extends Follmer's approach, incorporating rough path concepts and invariance conditions.

Findings

Defines pathwise stochastic integrals as limits of Riemann sums.

Shows integrals coincide with rough path integrals.

Identifies conditions for invariance of quadratic variation and L9vy area.

Abstract

We develop a general framework for pathwise stochastic integration that extends F\"ollmer's classical approach beyond gradient-type integrands and standard left-point Riemann sums and provides pathwise counterparts of It\^o, Stratonovich, and backward It\^o integration. More precisely, for a continuous path admitting both quadratic variation and L\'evy area along a fixed sequence of partitions, we define pathwise stochastic integrals as limits of general Riemann sums and prove that they coincide with integrals defined with respect to suitable rough paths. Furthermore, we identify necessary and sufficient conditions under which the quadratic variation and the L\'evy area of a continuous path are invariant with respect to the choice of partition sequences.