It\^{o}--F\"{o}llmer Calculus in Banach Spaces I: The It\^{o} Formula

TL;DR

This paper extends F"{o}llmer's pathwise It ext{"o} formula to Banach space-valued cadlag paths, relaxing partition assumptions for quadratic variation, advancing stochastic calculus in infinite-dimensional spaces.

Contribution

It introduces a generalized It ext{"o} formula for Banach space-valued paths and relaxes partition conditions for quadratic variation, broadening applicability.

Findings

Proves F"{o}llmer's It ext{"o} formula in Banach spaces.

Relaxes assumptions on partition sequences for quadratic variation.

Enhances stochastic calculus tools in infinite-dimensional analysis.

Abstract

We prove F\"{o}llmer's pathwise It\^{o} formula for a Banach space-valued c\`{a}dl\`{a}g path. We also relax the assumption on the sequence of partitions along which we treat the quadratic variation of a path.