It\^o formula for planarly branched rough paths
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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 $\frac{1}{4}< \alpha \leq \frac{1}{2}$.
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Jump It\^o-type formula with arbitrary regularity
A fully pathwise Itô formula is derived for finite p-variation paths with jumps, separating the reduced rough integral from explicit left- and right-jump correction terms.
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