It\^o formula for planarly branched rough paths

TL;DR

This paper extends the Itô formula to planarly branched rough paths with roughness between 1/4 and 1/2, broadening the applicability of stochastic calculus in rough path theory.

Contribution

It proves the Itô formula for planarly branched rough paths with roughness 1/4 < α ≤ 1/2, expanding the class of processes where stochastic calculus applies.

Findings

Itô formula established for planarly branched rough paths with roughness 1/4 < α ≤ 1/2.

Broader class of stochastic processes covered by the new Itô formula.

Enhances the theoretical framework of rough path analysis.

Abstract

The It\^o formula, originated by K. It\^o, is focus on the stochastic calculus, where many stochastic processes can be placed under the framework of rough paths. In rough path theory, It\^o formulas have been proved for rough paths with roughness $\frac{1}{3}< \alpha \leq \frac{1}{2}$ and branched rough paths with roughness $0< \alpha \leq 1$. Planarly branched rough paths contain more random processes than rough paths and branched rough paths. In the present paper, we prove the It\^o formula for planarly branched rough paths with roughness $\frac{1}{4}< \alpha \leq \frac{1}{2}$.