Jump Itô-type formula with arbitrary regularity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915970611675136 |
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| author | Li, Nannan Gao, Xing |
| author_facet | Li, Nannan Gao, Xing |
| contents | We establish an Itô-type formula for finite $p$-variation paths with jumps for arbitrary $p\geq 1$. The formula is stated in a fully pathwise form and separates the reduced rough integral from explicit left- and right-jump correction terms. In the càdlàg case, only the left-jump correction remains, while in the continuous case, both jump correction terms vanish and the formula recovers the corresponding continuous arbitrary-regularity change-of-variable formula. The proof is based on the reduced rough path framework and a refinement Riemann-Stieltjes convergence criterion adapted to discontinuous paths. This approach allows us to handle the higher-order Taylor expansions required for large values of $p$ and to control the interaction between rough increments and discrete jumps.
As applications, we derive Itô-type formulas for stochastic processes whose sample paths have finite $p$-variation, including pure-jump models and mixed fractional Brownian-jump signals. The latter class includes cases with Hurst parameter $H\leq 1/3$, which fall outside the regime $2\leq p<3$. We also obtain chain-rule identities for nonlinear observables of càdlàg finite-$p$-variation solutions of random differential equations with jumps, together with a pathwise log-wealth decomposition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_27627 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Jump Itô-type formula with arbitrary regularity Li, Nannan Gao, Xing Probability 60L20, 60H99, 60L90 We establish an Itô-type formula for finite $p$-variation paths with jumps for arbitrary $p\geq 1$. The formula is stated in a fully pathwise form and separates the reduced rough integral from explicit left- and right-jump correction terms. In the càdlàg case, only the left-jump correction remains, while in the continuous case, both jump correction terms vanish and the formula recovers the corresponding continuous arbitrary-regularity change-of-variable formula. The proof is based on the reduced rough path framework and a refinement Riemann-Stieltjes convergence criterion adapted to discontinuous paths. This approach allows us to handle the higher-order Taylor expansions required for large values of $p$ and to control the interaction between rough increments and discrete jumps. As applications, we derive Itô-type formulas for stochastic processes whose sample paths have finite $p$-variation, including pure-jump models and mixed fractional Brownian-jump signals. The latter class includes cases with Hurst parameter $H\leq 1/3$, which fall outside the regime $2\leq p<3$. We also obtain chain-rule identities for nonlinear observables of càdlàg finite-$p$-variation solutions of random differential equations with jumps, together with a pathwise log-wealth decomposition. |
| title | Jump Itô-type formula with arbitrary regularity |
| topic | Probability 60L20, 60H99, 60L90 |
| url | https://arxiv.org/abs/2604.27627 |