It\^o formula for reduced rough paths

Abstract

The It\^o formula, also known as the change-of-variables formula, is a cornerstone of It\^o stochastic calculus. Over time, this formula has been extended to apply to random processes for which classical calculus is insufficient. Since every random process exhibits some degree of regularity, rough path theory provides a natural framework for treating them uniformly. In this paper, we extend the It\^o formula for reduced rough paths, broadening the range of roughness from the previously known case $\frac{1}{3} < \alpha \leq \frac{1}{2}$ to the more singular regime $\frac{1}{4} < \alpha \leq \frac{1}{3}$.