Jump It\^o-type formula with arbitrary regularity

We establish an It\^o-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 c\`adl\`ag 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 It\^o-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 c\`adl\`ag finite-$p$-variation solutions of random differential equations with jumps, together with a pathwise log-wealth decomposition.