Rough Functional It\^o Formula

TL;DR

This paper establishes a rough Itô formula for path-dependent functionals of Hölder continuous paths, extending stochastic calculus tools to rough path settings with new analytical techniques.

Contribution

It introduces a novel rough Itô formula for path-dependent functionals using sewing lemma and Taylor approximation, broadening the scope of stochastic calculus for rough paths.

Findings

Proves a rough Itô formula for α-Hölder paths

Combines sewing lemma with path-dependent derivatives

Extends stochastic calculus to rough path analysis

Abstract

We prove a rough It\^o formula for path-dependent functionals of $\alpha$-H\"older continuous paths for $\alpha\in(0,1)$. Our approach combines the sewing lemma and a Taylor approximation in terms of path-dependent derivatives.