Jump It\^o-type formula with arbitrary regularity

Nannan Li; Xing Gao

arxiv: 2604.27627 · v1 · submitted 2026-04-30 · 🧮 math.PR

Jump It\^o-type formula with arbitrary regularity

Nannan Li , Xing Gao This is my paper

Pith reviewed 2026-05-07 07:18 UTC · model grok-4.3

classification 🧮 math.PR

keywords Itô formulap-variationrough pathsjumpscàdlàg pathsstochastic calculuschain rulefinite variation

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The pith

Finite p-variation paths with jumps obey a pathwise Itô formula that separates the reduced rough integral from explicit left- and right-jump corrections for any p ≥ 1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an Itô-type formula that holds for paths of finite p-variation with jumps, valid for arbitrary p greater than or equal to 1. The formula is fully pathwise and distinguishes the contribution of the continuous rough part, captured by a reduced rough integral, from explicit correction terms for left and right jumps. A sympathetic reader would care because this covers a wider class of stochastic processes than previous results, including pure-jump models and combinations with fractional Brownian motion of low Hurst index. In the continuous case the corrections disappear and the formula reduces to the known change-of-variable result for arbitrary regularity.

Core claim

We establish an Itô-type formula for finite p-variation paths with jumps for arbitrary p≥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.

What carries the argument

The reduced rough path framework with a refined Riemann-Stieltjes convergence criterion for discontinuous paths that handles higher-order Taylor expansions around jumps.

Load-bearing premise

The refined Riemann-Stieltjes convergence criterion adapted to discontinuous paths must hold in order to control the interaction between rough increments and discrete jumps during higher-order Taylor expansions.

What would settle it

Explicit computation of the change in a test function along a specific càdlàg path of finite p-variation with one jump, showing that the equality fails when the left-jump correction is included or omitted, would falsify the formula.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends the pathwise Itô formula to jumps at any p-variation level via reduced rough paths and a refined Riemann-Stieltjes criterion, but the higher-order remainder control across jumps for p>2 is the step that needs the closest look.

read the letter

The main result is a pathwise change-of-variable formula for finite p-variation paths that includes jumps, written so the reduced rough integral sits separately from explicit left- and right-jump correction terms. For cadlag paths only the left correction survives, and the continuous case drops both corrections and recovers the known arbitrary-regularity formula. That separation is clean and the applications to mixed fractional Brownian-jump signals with H ≤ 1/3 and to chain rules for nonlinear observables on jump RDEs are the parts that could actually get used. The proof strategy builds on the reduced rough path framework with an adaptation of the Riemann-Stieltjes convergence criterion to discontinuous paths, which is the natural way to try this. The paper ships the formula, the proof outline, and the applications in one package, which is helpful. The soft spot is the refined convergence criterion itself. For p > 2 the Taylor expansion of order [p] produces second- and higher-order increments that interact with both the continuous rough parts and the discrete jumps. The abstract claims the adaptation controls the remainders uniformly, but the estimates need to be checked for uniformity in jump size and for whether any extra logarithmic factors or restrictions on the jump locations appear. If those bounds are stated explicitly and hold without hidden conditions, the argument goes through; otherwise the separation into rough integral plus corrections may only be justified in restricted regimes. This is for readers already comfortable with rough path theory who need formulas that reach low-regularity jump processes. A specialist in pathwise stochastic analysis or jump-driven models would find the applications section worth reading. It is worth sending to a referee because the claim is concrete, the framework is standard, and the new technical step is isolated enough that a careful check can settle it.

Referee Report

1 major / 2 minor

Summary. The manuscript establishes a pathwise Itô-type formula for finite p-variation paths with jumps, valid for arbitrary p ≥ 1. The formula separates a reduced rough integral from explicit left- and right-jump correction terms. In the càdlàg setting only the left correction remains; in the continuous case both corrections vanish and the formula recovers the known continuous arbitrary-regularity change-of-variable result. The proof relies on the reduced rough path framework together with a refined Riemann-Stieltjes convergence criterion adapted to discontinuous paths. Applications include Itô formulas for pure-jump processes and mixed fractional Brownian-jump signals (including Hurst index H ≤ 1/3), chain-rule identities for nonlinear observables of càdlàg finite-p-variation solutions of random differential equations with jumps, and a pathwise log-wealth decomposition.

Significance. If the central derivation holds, the result meaningfully extends rough-path techniques to jump paths at arbitrary regularity, including regimes outside 2 ≤ p < 3. The explicit separation of the rough integral from jump corrections, the recovery of the continuous case, and the concrete applications to mixed fractional Brownian motion with jumps and to random differential equations constitute clear strengths. The pathwise formulation and absence of free parameters or fitted quantities are also positive features.

major comments (1)

[Proof of the main Itô formula (Section 4)] The refined Riemann-Stieltjes convergence criterion adapted to discontinuous paths (invoked in the proof of the main theorem to control Taylor remainders of order [p] for p > 2) is load-bearing. The manuscript must supply an explicit statement of the criterion together with uniform estimates showing that second- and third-order remainder terms vanish when interacting with both continuous rough increments and discrete jumps; without these estimates the separation into a reduced rough integral plus explicit left/right corrections is not justified.

minor comments (2)

[Abstract] Abstract, line 5: 'a refinement Riemann-Stieltjes' should read 'a refinement of the Riemann-Stieltjes'.
[Section 2] Notation for left- and right-jump corrections should be introduced once and used consistently throughout the statements of the main theorem and the applications.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We are pleased that the referee recognizes the significance of extending rough-path techniques to jump paths at arbitrary regularity. Below we provide a point-by-point response to the major comment.

read point-by-point responses

Referee: [Proof of the main Itô formula (Section 4)] The refined Riemann-Stieltjes convergence criterion adapted to discontinuous paths (invoked in the proof of the main theorem to control Taylor remainders of order [p] for p > 2) is load-bearing. The manuscript must supply an explicit statement of the criterion together with uniform estimates showing that second- and third-order remainder terms vanish when interacting with both continuous rough increments and discrete jumps; without these estimates the separation into a reduced rough integral plus explicit left/right corrections is not justified.

Authors: We agree with the referee that the refined Riemann-Stieltjes convergence criterion is load-bearing for the proof and that its explicit statement with the necessary estimates is essential for justifying the separation of the reduced rough integral from the jump correction terms. Upon reviewing the manuscript, we acknowledge that while the criterion is invoked and some estimates are provided in the appendix, a consolidated explicit statement is not present in a single location. In the revised manuscript, we will add a new lemma in Section 3 that clearly states the refined convergence criterion adapted to discontinuous paths, together with uniform bounds on the second- and third-order Taylor remainders. These bounds will explicitly demonstrate that the remainders vanish in the limit when interacting with continuous rough increments (controlled via the reduced rough path) and discrete jumps (controlled via the finite p-variation). This revision will make the proof self-contained and transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on external reduced rough path framework plus new refinement for jumps.

full rationale

The paper claims an Itô-type formula derived from the reduced rough path framework together with a newly introduced refinement of the Riemann-Stieltjes convergence criterion adapted to discontinuous paths. This refinement is used to control higher-order Taylor remainders and jump-rough interactions, and the resulting formula is shown to recover the known continuous case when jumps vanish. No step reduces the target formula to a tautological redefinition of its own inputs, a fitted parameter renamed as a prediction, or a load-bearing uniqueness claim imported solely from the authors' prior unverified work. The central estimates are presented as independent adaptations rather than self-referential constructions, and the result is benchmarked against external limits (continuous arbitrary-regularity formulas and applications to mixed fractional Brownian-jump processes).

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The claim rests on standard assumptions from rough path theory adapted to the discontinuous setting, with no free parameters or new entities introduced.

axioms (3)

domain assumption The paths under consideration have finite p-variation for some p ≥ 1
This regularity condition is fundamental to defining the rough path lift and the formula.
domain assumption A reduced rough path framework can be applied to paths with jumps
The proof is based on this framework as stated in the abstract.
ad hoc to paper The refined Riemann-Stieltjes convergence criterion holds for discontinuous paths
This is adapted in the paper to handle the interaction between rough increments and jumps.

pith-pipeline@v0.9.0 · 5519 in / 1549 out tokens · 119563 ms · 2026-05-07T07:18:44.275934+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 1 canonical work pages

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Hairer and D

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[1] [1]

Bellingeri, Quasi-geometric rough paths and rough change of variable formula, Ann

C. Bellingeri, Quasi-geometric rough paths and rough change of variable formula, Ann. Inst. Henri Poincar ´e Probab. Stat.,59(3)(2023), 1398-1433. 2

2023

[2] [2]

Bellingeri, E

C. Bellingeri, E. Ferrucci and N. Tapia, Branched It ˆo formula and natural Itˆo-Stratonovich isomorphism, Adv. Math.,484(2026), Paper No. 110687. 2

2026

[3] [3]

Cont and N

R. Cont and N. Perkowski, Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity, Trans. Amer. Math. Soc. Ser. B,6(2019), 161-186. 2, 3, 4, 6, 7, 10

2019

[4] [4]

P. K. Friz and M. Hairer, A course on rough paths, Springer, 2020. 2, 17

2020

[5] [5]

P. K. Friz and H. Zhang, Differential equations driven by rough paths with jumps, J. Differential Equations, 264(10)(2018), 6226-6301. 2, 3, 4, 8, 9, 17, 18

2018

[6] [6]

Hairer and D

M. Hairer and D. Kelly, Geometric versus non-geometric rough paths, Ann. Inst. Henri Poincar´e Probab. Stat., 51(1)(2015), 207-251. 2

2015

[7] [7]

T. H. Hildebrandt, Definitions of Stieltjes integrals of the Riemann type, Amer. Math. Monthly,45(5)(1938), 265-278. 3, 8

1938

[8] [8]

T. H. Hildebrandt, Introduction to the Theory of Integration, Academic Press, New York, 1963. 3, 8

1963

[9] [9]

It ˆo, On a formula concerning stochastic differentials, Nagoya Math

K. It ˆo, On a formula concerning stochastic differentials, Nagoya Math. J.,3(1951), 55-65. 2

1951

[10] [10]

Kelly, It ˆo corrections in stochastic equations, Ph.D

D. Kelly, It ˆo corrections in stochastic equations, Ph.D. thesis, University of Warwick, 2012. 2

2012

[11] [11]

Li and X

N. Li and X. Gao, It ˆo formula for planarly branched rough paths, arXiv:2501.11886. 2

work page arXiv

[12] [12]

Li and X

N. Li and X. Gao, It ˆo formula for reduced rough paths, Statist. Probab. Lett.,234(2026), Paper No. 110712. 2

2026

[13] [13]

T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana,14(2)(1998), 215-310. 2, 17

1998

[14] [14]

Sato, L ´evy processes and infinitely divisible distributions, Cambridge University Press, 1999

K. Sato, L ´evy processes and infinitely divisible distributions, Cambridge University Press, 1999. 16 School ofMathematics andStatistics, LanzhouUniversityLanzhou, 730000, China Email address:linn2024@lzu.edu.cn School ofMathematics andStatistics, LanzhouUniversityLanzhou, 730000, China; GansuProvincialRe- searchCenter forBasicDisciplines ofMathematics a...

1999