arxiv: 2605.11783 · v1 · submitted 2026-05-12 · 🧮 math.PR

Recognition: 2 theorem links

· Lean Theorem

Stability of Compensated Jump Integrals under Quadratic Variation Convergence

Philip Kennerberg

Authors on Pith no claims yet

Pith reviewed 2026-05-13 05:20 UTC · model grok-4.3

classification 🧮 math.PR

keywords quadratic variation convergencecompensated jump integralsucp convergencecadlag processesjump measuresforbidden bands principlecompensator mass control

0 comments

The pith

Quadratic variation convergence alone implies ucp stability of compensated jump integrals under local linear growth on the integrands.

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

The paper establishes that if the quadratic variation difference [X^n - X]_t converges to zero in probability, then compensated integrals of the form integral f_n d(mu_n - nu_n) converge uniformly on compacts in probability. This holds when the integrands satisfy local linear growth and converge locally uniformly, without requiring semimartingale convergence, characteristic convergence, or tightness. A reader cares because the result reveals that quadratic variation imposes strong structural control on the organization of jumps in cadlag processes, via two mechanisms: a forbidden bands principle that blocks jumps from crossing moving thresholds, and compensator mass control that aligns large jumps with their compensators. The findings apply to general cadlag processes without assuming independence, stationarity, or Markovianity.

Core claim

Under the assumption that [X^n - X]_t to 0 in probability, the compensated jump integrals converge ucp when the integrands f_n satisfy local linear growth and locally uniform convergence. The proof is based on a forbidden bands principle showing that quadratic variation convergence prevents jumps from crossing suitably chosen moving threshold regions, and a compensator mass control mechanism combining threshold-separated alignment of large predictable jumps with a counting argument for compensator atoms. The results require neither semimartingale convergence nor convergence of characteristics nor uniform tightness nor global structural assumptions.

What carries the argument

The forbidden bands principle together with compensator mass control, which together extract jump organization rigidity directly from quadratic variation convergence.

If this is right

Compensated jump integrals remain stable under quadratic variation convergence even without full convergence of the processes or their characteristics.
Large jumps in the compensators align in a threshold-separated manner controlled solely by the quadratic variation limit.
The jump measures mu_n and mu are forced into a rigid organization that prevents uncontrolled crossings of moving bands.
Stochastic integrals against jump measures inherit continuity properties from quadratic variation limits alone.
The framework applies to arbitrary cadlag processes without invoking independence, stationarity, or Markov assumptions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same mechanisms may extend stability results to other functionals of the jump measure, such as path-dependent integrals or supremum functionals.
Quadratic variation convergence could serve as a minimal condition for convergence in the Skorokhod topology restricted to jump behavior.
Numerical schemes for jump processes might be validated by checking quadratic variation closeness rather than full characteristic matching.
The rigidity identified here suggests quadratic variation as a candidate for a complete invariant in certain classes of cadlag semimartingales.

Load-bearing premise

The integrands satisfy local linear growth and locally uniform convergence so that quadratic variation convergence alone triggers the forbidden bands and compensator controls.

What would settle it

Construct cadlag processes X^n and X with [X^n - X]_t to 0 in probability but with an integrand satisfying the local growth conditions where the compensated integral fails to converge ucp.

read the original abstract

We study the stability of compensated jump integrals under convergence of quadratic variation alone. Let \(X\) and \(\{X^n\}_{n\ge1}\) be c\`adl\`ag processes with jump measures \(\mu,\mu_n\) and predictable compensators \(\nu,\nu_n\). Under the assumption \[ [X^n-X]_t \to 0 \qquad\text{in probability}, \] we establish ucp convergence of compensated jump integrals of the form \[ \int_0^. \int_{\mathbb R} f_n(s,x)(\mu_n-\nu_n)(ds,dx) \] under local linear growth and locally uniform convergence assumptions on the integrands. The proof is based on two structural mechanisms. The first is a forbidden bands principle, showing that quadratic variation convergence prevents jumps from crossing suitably chosen moving threshold regions. The second is a compensator mass control mechanism, which combines threshold-separated alignment of large predictable jumps with a counting argument for the associated compensator atoms. The results require neither semimartingale convergence, convergence of characteristics, uniform tightness, nor global structural assumptions such as independence, stationarity, or Markovianity. More broadly, they show that quadratic variation convergence imposes a substantially stronger rigidity on the jump organization of c\`adl\`ag processes than one might initially expect.

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

Quadratic variation convergence alone gives ucp stability for compensated jump integrals via forbidden bands and compensator mass control, a clean technical lemma with limited scope.

read the letter

The main point is that if quadratic variation converges in probability, then the compensated integrals converge ucp under local linear growth and local uniform convergence of the integrands. The paper gets this without semimartingale convergence, convergence of characteristics, or uniform tightness, which is the real advance here. The two mechanisms do the work: forbidden bands stop jumps from crossing chosen threshold regions because small quadratic variation prevents that, and compensator mass control aligns large predictable jumps with a counting argument on the atoms to bound the mass. These steps follow directly from the quadratic variation assumption plus the local conditions, and the stress-test confirms no hidden uniformity or circular steps are needed. The local setup keeps everything from requiring global assumptions like Markov or stationarity, which is a plus for generality within the cadlag setting. The soft spots are proportionate to the claim. The result stays narrow, so it functions mainly as a lemma rather than a broad tool, and applications will still need to verify the local conditions hold in each case. No internal gaps appear in the derivation from the structural lemmas to the final ucp statement. This is for people working on convergence theorems for stochastic integrals with jumps in mathematical probability. A reader who needs stability results under weak assumptions on the driving processes would find the techniques useful. It deserves a serious referee because the central mechanisms are grounded and the assumptions are stated clearly.

Referee Report

0 major / 3 minor

Summary. The paper claims that under the assumption [X^n - X]_t → 0 in probability, the compensated jump integrals ∫_0^. ∫_R f_n(s,x)(μ_n - ν_n)(ds,dx) converge ucp, assuming only local linear growth and locally uniform convergence of the integrands f_n. The proof introduces two structural mechanisms—a forbidden bands principle preventing jumps from crossing moving thresholds and a compensator mass control via threshold-separated alignment and counting arguments for compensator atoms—without requiring semimartingale convergence, convergence of characteristics, uniform tightness, or global assumptions such as independence or Markovianity.

Significance. If the result holds, it is significant for showing that quadratic variation convergence alone enforces strong rigidity on the jump organization of càdlàg processes, enabling stability of compensated integrals under minimal local conditions. This is a parameter-free derivation relying on directly derived structural mechanisms rather than fitted parameters or additional hypotheses, which enhances its utility in stochastic processes theory where only QV convergence is available.

minor comments (3)

[Abstract] The abstract states ucp convergence but does not explicitly identify the limit process (the compensated integral w.r.t. (μ - ν)); adding this would improve precision.
[Introduction] The notation for jump measures μ_n, μ and compensators ν_n, ν is introduced in the abstract but would benefit from an early dedicated notation paragraph in the introduction for readers unfamiliar with the setup.
[Assumptions section] The locally uniform convergence assumption on f_n is used to localize the ucp result; a brief illustrative example of functions satisfying this (e.g., truncated indicators) would clarify its applicability without altering the proof.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition of its significance for stochastic processes theory under minimal assumptions. The recommendation is for minor revision, yet no specific major comments or points requiring clarification were raised in the report. We have re-examined the paper and confirm that the stated results and proofs stand as presented.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from quadratic variation assumption

full rationale

The paper establishes ucp convergence of the compensated integrals directly from the given assumption [X^n - X]_t → 0 in probability together with local linear growth and locally uniform convergence of the integrands. The forbidden bands principle and compensator mass control are derived as structural consequences of quadratic variation convergence via counting arguments and threshold separation, without any reduction to fitted parameters, self-definitional equivalences, or load-bearing self-citations. The central claim remains independent of its inputs and does not rename known results or smuggle ansatzes via prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard properties of cadlag processes, quadratic variation, and jump measures from stochastic calculus. The two structural mechanisms (forbidden bands and compensator mass control) are introduced as proof techniques rather than new axioms or entities.

axioms (1)

standard math Standard properties of cadlag processes, quadratic variation, and predictable compensators for jump measures
Invoked throughout the abstract as background for the convergence statements.

pith-pipeline@v0.9.0 · 5523 in / 1256 out tokens · 33060 ms · 2026-05-13T05:20:13.124028+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

quadratic variation convergence imposes a substantially stronger rigidity on the jump organization

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. The Annals of Probability, 19(3), 1035--1070

work page 1991
[2]

M\'emin, J. (1980). Espaces de semi martingales et changement de probabilit\'e. Zeitschrift f\"ur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 52, 9--39

work page 1980
[3]

Liptser, R. S. and Shiryaev, A. N. (1989). Theory of Martingales. Kluwer Academic Publishers

work page 1989
[4]

and Wiktorsson, M

Kennerberg, P. and Wiktorsson, M. (2026). Stability in quadratic variation. Collectanea Mathematica

work page 2026
[5]

Meyer, P.-A. (1966). Probabilit\'es et Potentiel. Hermann, Paris

work page 1966
[6]

and Shiryaev, A

Jacod, J. and Shiryaev, A. (2003). Limit Theorems for Stochastic Processes. Second edition. Springer

work page 2003
[7]

and Yan, J

He, S., Wang, J. and Yan, J. (1992). Semimartingale Theory and Stochastic Calculus. Science Press and CRC Press

work page 1992
[8]

and Watanabe, S

Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes. Second edition. North-Holland

work page 1989
[9]

Protter, P. (2005). Stochastic Integration and Differential Equations. Second edition. Springer

work page 2005