Itô formula for planarly branched rough paths (2501.11886v2)

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 $\frac{1}{4}< \alpha \leq \frac{1}{2}$.