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arxiv: 2501.11886 · v2 · pith:MW6YKGHY · submitted 2025-01-21 · math.PR · math.CA

It\^o formula for planarly branched rough paths

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classification math.PR math.CA
keywords roughpathsbranchedfracalphaformulaplanarlyroughness
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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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  1. Jump It\^o-type formula with arbitrary regularity

    math.PR 2026-04 unverdicted novelty 7.0

    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.