Itô formula for planarly branched rough paths
Nannan Li, Xing Gao
TL;DR
The paper advances rough path theory by establishing an Itô formula for planarly branched rough paths with roughness $\alpha$ in $\big(\tfrac{1}{4}, \tfrac{1}{2}\big]$, leveraging the Munthe-Kaas-Wright Hopf algebra. It introduces and analyzes ${\bf X}$-controlled planarly branched rough paths, defines rough integrals in two $\alpha$-regimes, and demonstrates that smooth compositions yield new controlled paths, enabling a Taylor-type expansion for rough differential equations. The main contributions are the Itô formulas for the simple case $F(X)$ and the general case $F(Y)$, with truncated expansions at $N=2$ (for $\alpha>\tfrac{1}{3}$) and $N=3$ (for $\alpha\in(\tfrac{1}{4}, \tfrac{1}{3}]$), plus bracket extensions and a discussion of challenges for $\alpha\le \tfrac{1}{4}$. These results broaden the applicability of stochastic calculus within the planar-branching framework and pave the way for further analysis of planarly branched rough differential equations.
Abstract
The Itô formula, originated by K. Itô, is focus on the stochastic calculus, where many stochastic processes can be placed under the framework of rough paths. In rough path theory, Itô formulas have been proved for rough paths with roughness $\frac{1}{3}< α\leq \frac{1}{2}$ and branched rough paths with roughness $0< α\leq 1$. Planarly branched rough paths contain more random processes than rough paths and branched rough paths. In the present paper, we prove the Itô formula for planarly branched rough paths with roughness $\frac{1}{4}< α\leq \frac{1}{2}$.