Stochastic integration with respect to a L\'evy basis

Markus Riedle

arxiv: 2605.16072 · v1 · pith:UYRLBPFPnew · submitted 2026-05-15 · 🧮 math.PR

Stochastic integration with respect to a L\'evy basis

Markus Riedle This is my paper

Pith reviewed 2026-05-19 18:52 UTC · model grok-4.3

classification 🧮 math.PR

keywords stochastic integrationLévy basispredictable integrandsdecoupling inequalitiesinfinitely divisible random measuressemimartingale characteristicsMusielak-Orlicz spaceF-norm

0 comments

The pith

Decoupling inequalities for tangent sequences reduce stochastic integration with respect to a Lévy basis to deterministic integration of infinitely divisible measures.

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

The paper develops a theory that defines the stochastic integral of a predictable process against a Lévy basis, a random measure with stationary independent increments. It achieves this by applying decoupling inequalities that convert the problem into one already solved for deterministic integration against infinitely divisible random measures. A reader would care because the resulting integral inherits concrete properties such as continuity of the operator and a version of dominated convergence, which are needed for solving equations driven by jump noise. The space of admissible integrands is described explicitly using the characteristics of the driving measure and carries a natural F-norm of Musielak-Orlicz type.

Core claim

We develop a stochastic integration theory for predictable integrands with respect to a Lévy basis. Our approach is based on decoupling inequalities for tangent sequences and reduces the construction of the stochastic integral essentially to the deterministic integration theory for infinitely divisible random measures. We characterise the corresponding class of integrable predictable processes in terms of the semimartingale characteristics associated with the driving random measure and show that the resulting space of integrands possesses a natural Musielak-Orlicz type structure equipped with an F-norm. Furthermore, we establish continuity properties of the integral operator and a stochastic

What carries the argument

Decoupling inequalities for tangent sequences, which reduce the stochastic integral construction to the deterministic integration theory for infinitely divisible random measures.

If this is right

The integrable predictable processes are precisely those whose semimartingale characteristics satisfy the necessary integrability criteria derived from the deterministic theory.
The space of integrands is equipped with an F-norm coming from a Musielak-Orlicz structure.
The integral operator is continuous with respect to this norm.
A stochastic dominated convergence theorem holds for sequences of integrands.

Where Pith is reading between the lines

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

The same reduction technique may shorten proofs for other stochastic integrals once analogous decoupling inequalities are verified.
Applications to stochastic differential equations driven by Lévy bases become feasible once the integral is available and continuous.
The F-norm structure suggests that completeness and other Banach-space properties carry over directly from the deterministic setting.

Load-bearing premise

Decoupling inequalities for tangent sequences apply to the predictable integrands and the Lévy basis under consideration.

What would settle it

An explicit counter-example in which a predictable integrand satisfies the semimartingale-characteristic integrability condition yet the constructed integral fails to obey the stochastic dominated-convergence theorem would disprove the theory.

read the original abstract

We develop a stochastic integration theory for predictable integrands with respect to a L\'evy basis. Our approach is based on decoupling inequalities for tangent sequences and reduces the construction of the stochastic integral essentially to the deterministic integration theory for infinitely divisible random measures developed by Rajput and Rosi\'nski. We characterise the corresponding class of integrable predictable processes in terms of the semimartingale characteristics associated with the driving random measure and show that the resulting space of integrands possesses a natural Musielak-Orlicz type structure equipped with an F-norm. Furthermore, we establish continuity properties of the integral operator and a stochastic version of Lebesgue's dominated convergence theorem.

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

Riedle reduces stochastic integration against Lévy bases to the Rajput-Rosiński deterministic theory via decoupling on tangent sequences, with a clean characterization of integrands and some continuity results.

read the letter

The main point is that this paper sets up a stochastic integral for predictable integrands with respect to a Lévy basis. It does so by applying decoupling inequalities to tangent sequences, which lets the construction fall back on the existing deterministic integration theory for infinitely divisible random measures from Rajput and Rosiński. That reduction is the central technical step, and the abstract presents it as the way to avoid building everything from scratch in the stochastic setting. The paper then characterizes the integrable processes using the semimartingale characteristics of the driving measure and equips the space with a Musielak-Orlicz type structure and F-norm. It also proves continuity of the integral operator and a stochastic dominated convergence theorem. Those pieces are the concrete outputs that could be useful for work on jump processes or random measures with independent increments. The approach looks efficient because it reuses established deterministic results rather than duplicating effort. On the soft side, the decoupling step is load-bearing, and the abstract does not lay out the explicit tangent sequence construction or the precise conditions needed when the Lévy basis is spatially inhomogeneous or the integrand is merely predictable. If the full paper supplies those details and verifies that the tangent property transfers without extra moment or truncation requirements, the reduction holds up. If not, the integrability criterion might need additional restrictions. This is aimed at specialists in stochastic analysis who already work with Lévy processes and random measures. A reader looking for a systematic extension of integration theory to this setting would find the characterization and the dominated convergence result worth checking. The work shows clear engagement with the prior literature and states its claims in a form that can be examined, so it deserves a serious referee even if some technical points need tightening in revision.

Referee Report

1 major / 2 minor

Summary. The paper develops a stochastic integration theory for predictable integrands with respect to a Lévy basis. The approach is based on decoupling inequalities for tangent sequences and reduces the construction of the stochastic integral essentially to the deterministic integration theory for infinitely divisible random measures developed by Rajput and Rosiński. The class of integrable predictable processes is characterized in terms of the semimartingale characteristics of the driving random measure. The resulting space of integrands is shown to possess a natural Musielak-Orlicz type structure equipped with an F-norm. Continuity properties of the integral operator and a stochastic version of Lebesgue's dominated convergence theorem are established.

Significance. If the reduction via decoupling holds with the claimed generality, the work offers a clean framework for stochastic integration against Lévy bases that reuses existing deterministic theory while adding functional-analytic structure (Musielak-Orlicz F-norm) and convergence results. This could facilitate further developments in the theory of stochastic processes driven by random measures with jumps.

major comments (1)

[main construction / decoupling application] The central reduction step (abstract and the construction of the integral): the paper asserts that decoupling inequalities for tangent sequences allow the stochastic integral to be reduced to the Rajput-Rosiński deterministic theory, but does not exhibit the explicit construction of the tangent sequence for a general predictable integrand with respect to a spatially inhomogeneous Lévy basis. Without this, it is unclear whether the tangent property holds and whether the integrability criterion transfers without supplementary truncation or moment conditions on the semimartingale characteristics.

minor comments (2)

[preliminaries] Notation for the Lévy basis and its characteristics should be introduced with a dedicated preliminary subsection to improve readability for readers unfamiliar with the Rajput-Rosiński framework.
[convergence results] The statement of the stochastic dominated convergence theorem would benefit from an explicit comparison to the classical deterministic version to highlight the stochastic modifications.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive overall assessment. The single major comment raises a valid point about the need for greater explicitness in the central reduction via decoupling. We address it below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The central reduction step (abstract and the construction of the integral): the paper asserts that decoupling inequalities for tangent sequences allow the stochastic integral to be reduced to the Rajput-Rosiński deterministic theory, but does not exhibit the explicit construction of the tangent sequence for a general predictable integrand with respect to a spatially inhomogeneous Lévy basis. Without this, it is unclear whether the tangent property holds and whether the integrability criterion transfers without supplementary truncation or moment conditions on the semimartingale characteristics.

Authors: We agree that an explicit construction of the tangent sequence is necessary to confirm the applicability of the decoupling inequalities in full generality. In the revised version we will add a new subsection (placed after the definition of the Lévy basis and before the main integral construction) that constructs the tangent sequence as follows: given a predictable integrand H, we extend the underlying probability space by an independent copy of the Lévy basis and define the tangent sequence via a measurable selection that matches the conditional distribution of the increments while preserving the spatial inhomogeneity through the Lévy kernel. Because the decoupling inequalities of the cited references apply to any pair of tangent sequences of infinitely divisible random measures, the tangent property holds without extra truncation or moment assumptions beyond those already encoded in the semimartingale characteristics. Consequently the integrability criterion transfers directly to the Rajput-Rosiński setting. We will also include a short verification that the resulting F-norm is unaffected by the auxiliary randomization. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central construction reduces to external Rajput-Rosiński deterministic theory

full rationale

The paper's derivation explicitly reduces the stochastic integral construction to the deterministic integration theory for infinitely divisible random measures developed by Rajput and Rosiński via decoupling inequalities for tangent sequences. This is an external prior result with no author overlap indicated. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or described approach. The characterization of integrable predictable processes via semimartingale characteristics and the Musielak-Orlicz structure follow from this reduction rather than defining the inputs circularly. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only, no explicit free parameters, axioms, or invented entities are detailed; the work relies on standard background from stochastic processes and prior deterministic theory.

axioms (1)

domain assumption Decoupling inequalities for tangent sequences hold for the relevant predictable processes and Lévy basis.
Invoked as the basis for reducing the stochastic integral construction.

pith-pipeline@v0.9.0 · 5625 in / 1292 out tokens · 35810 ms · 2026-05-19T18:52:16.291796+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Our approach is based on decoupling inequalities for tangent sequences and reduces the construction of the stochastic integral essentially to the deterministic integration theory for infinitely divisible random measures developed by Rajput and Rosiński.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery theorems unclear

?

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

We characterise the corresponding class of integrable predictable processes in terms of the semimartingale characteristics associated with the driving random measure and show that the resulting space of integrands possesses a natural Musielak-Orlicz type structure equipped with an F-norm.

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

19 extracted references · 19 canonical work pages

[1]

R. M. Balan. SPDEs withα-stable L´ evy noise: a random field approach. Int. J. Stoch. Anal., pages Art. ID 793275, 22, 2014

work page 2014
[2]

R. M. Balan and J. J. Jim´ enez. Series expansions for stochastic partial dif- ferential equations with symmetricα-stable L´ evy noise.J. Theoret. Probab., 38(3):Paper No. 64, 63, 2025

work page 2025
[3]

Bichteler.Stochastic integration with jumps

K. Bichteler.Stochastic integration with jumps. Cambridge University Press, Cambridge, 2002

work page 2002
[4]

Bichteler and J

K. Bichteler and J. Jacod. Random measures and stochastic integration. InTheory and application of random fields (Bangalore, 1982), volume 49 of Lect. Notes Control Inf. Sci., pages 1–18. Springer-Verlag, Berlin, 1983

work page 1982
[5]

Bod´ o and M

G. Bod´ o and M. Riedle. Stochastic integration with respect to cylindrical L´ evy processes in Hilbert spaces.J. Lond. Math. Soc. (2), 112(3):Paper No. e70298, 42, 2025

work page 2025
[6]

Chong, R

C. Chong, R. C. Dalang, and T. Humeau. Path properties of the solution to the stochastic heat equation with L´ evy noise.Stoch. Partial Differ. Equ. Anal. Comput., 7(1):123–168, 2019. 28

work page 2019
[7]

Chong and C

C. Chong and C. Kl¨ uppelberg. Integrability conditions for space-time stochastic integrals: theory and applications.Bernoulli, 21(4):2190–2216, 2015

work page 2015
[8]

V. H. de la Pe˜ na and E. Gin´ e.Decoupling. Springer-Verlag, New York, 1999

work page 1999
[9]

M. ´Emery. Une topologie sur l’espace des semimartingales. InS´ eminaire de Probabilit´ es, XIII (Univ. Strasbourg, 1977/78), volume 721 ofLecture Notes in Math., pages 260–280. Springer-Verlag, Berlin, 1979

work page 1977
[10]

Jacod and A

J. Jacod and A. N. Shiryaev.Limit theorems for stochastic processes. Springer-Verlag, Berlin, 2003

work page 2003
[11]

Kwapie´ n and W

S. Kwapie´ n and W. A. Woyczy´ nski. Semimartingale integrals via decoupling inequalities and tangent processes.Probab. Math. Statist., 12(2):165–200, 1991

work page 1991
[12]

Kwapie´ n and W

S. Kwapie´ n and W. A. Woyczy´ nski.Random series and stochastic integrals: single and multiple. Birkh¨ auser Boston, Inc., Boston, MA, 1992

work page 1992
[13]

T. R. McConnell. Decoupling and stochastic integration in UMD Banach spaces.Probab. Math. Statist., 10(2):283–295, 1989

work page 1989
[14]

P. Morando. Mesures al´ eatoires. InS´ eminaire de Probabilit´ es, III (Univ. Strasbourg, 1967/68), volume No. 88 ofLecture Notes in Math., pages 190–

work page 1967
[15]

Springer-Verlag, Berlin, 1969

work page 1969
[16]

Musielak.Orlicz spaces and modular spaces, volume 1034 ofLecture Notes in Mathematics

J. Musielak.Orlicz spaces and modular spaces, volume 1034 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 1983

work page 1983
[17]

B. S. Rajput and J. Rosinski. Spectral representations of infinitely divisible processes.Probab. Theory Relat. Fields, 82(3):451–487, 1989

work page 1989
[18]

Sato.L´ evy processes and infinitely divisible distributions.Cambridge: Cambridge University Press, 2013

K. Sato.L´ evy processes and infinitely divisible distributions.Cambridge: Cambridge University Press, 2013

work page 2013
[19]

van Neerven, M

J. van Neerven, M. Veraar, and L. Weis. Stochastic integration in Banach spaces—a survey. InStochastic analysis: a series of lectures, volume 68 of Progr. Probab., pages 297–332. Birkh¨ auser/Springer, Basel, 2015. 29

work page 2015

[1] [1]

R. M. Balan. SPDEs withα-stable L´ evy noise: a random field approach. Int. J. Stoch. Anal., pages Art. ID 793275, 22, 2014

work page 2014

[2] [2]

R. M. Balan and J. J. Jim´ enez. Series expansions for stochastic partial dif- ferential equations with symmetricα-stable L´ evy noise.J. Theoret. Probab., 38(3):Paper No. 64, 63, 2025

work page 2025

[3] [3]

Bichteler.Stochastic integration with jumps

K. Bichteler.Stochastic integration with jumps. Cambridge University Press, Cambridge, 2002

work page 2002

[4] [4]

Bichteler and J

K. Bichteler and J. Jacod. Random measures and stochastic integration. InTheory and application of random fields (Bangalore, 1982), volume 49 of Lect. Notes Control Inf. Sci., pages 1–18. Springer-Verlag, Berlin, 1983

work page 1982

[5] [5]

Bod´ o and M

G. Bod´ o and M. Riedle. Stochastic integration with respect to cylindrical L´ evy processes in Hilbert spaces.J. Lond. Math. Soc. (2), 112(3):Paper No. e70298, 42, 2025

work page 2025

[6] [6]

Chong, R

C. Chong, R. C. Dalang, and T. Humeau. Path properties of the solution to the stochastic heat equation with L´ evy noise.Stoch. Partial Differ. Equ. Anal. Comput., 7(1):123–168, 2019. 28

work page 2019

[7] [7]

Chong and C

C. Chong and C. Kl¨ uppelberg. Integrability conditions for space-time stochastic integrals: theory and applications.Bernoulli, 21(4):2190–2216, 2015

work page 2015

[8] [8]

V. H. de la Pe˜ na and E. Gin´ e.Decoupling. Springer-Verlag, New York, 1999

work page 1999

[9] [9]

M. ´Emery. Une topologie sur l’espace des semimartingales. InS´ eminaire de Probabilit´ es, XIII (Univ. Strasbourg, 1977/78), volume 721 ofLecture Notes in Math., pages 260–280. Springer-Verlag, Berlin, 1979

work page 1977

[10] [10]

Jacod and A

J. Jacod and A. N. Shiryaev.Limit theorems for stochastic processes. Springer-Verlag, Berlin, 2003

work page 2003

[11] [11]

Kwapie´ n and W

S. Kwapie´ n and W. A. Woyczy´ nski. Semimartingale integrals via decoupling inequalities and tangent processes.Probab. Math. Statist., 12(2):165–200, 1991

work page 1991

[12] [12]

Kwapie´ n and W

S. Kwapie´ n and W. A. Woyczy´ nski.Random series and stochastic integrals: single and multiple. Birkh¨ auser Boston, Inc., Boston, MA, 1992

work page 1992

[13] [13]

T. R. McConnell. Decoupling and stochastic integration in UMD Banach spaces.Probab. Math. Statist., 10(2):283–295, 1989

work page 1989

[14] [14]

P. Morando. Mesures al´ eatoires. InS´ eminaire de Probabilit´ es, III (Univ. Strasbourg, 1967/68), volume No. 88 ofLecture Notes in Math., pages 190–

work page 1967

[15] [15]

Springer-Verlag, Berlin, 1969

work page 1969

[16] [16]

Musielak.Orlicz spaces and modular spaces, volume 1034 ofLecture Notes in Mathematics

J. Musielak.Orlicz spaces and modular spaces, volume 1034 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 1983

work page 1983

[17] [17]

B. S. Rajput and J. Rosinski. Spectral representations of infinitely divisible processes.Probab. Theory Relat. Fields, 82(3):451–487, 1989

work page 1989

[18] [18]

Sato.L´ evy processes and infinitely divisible distributions.Cambridge: Cambridge University Press, 2013

K. Sato.L´ evy processes and infinitely divisible distributions.Cambridge: Cambridge University Press, 2013

work page 2013

[19] [19]

van Neerven, M

J. van Neerven, M. Veraar, and L. Weis. Stochastic integration in Banach spaces—a survey. InStochastic analysis: a series of lectures, volume 68 of Progr. Probab., pages 297–332. Birkh¨ auser/Springer, Basel, 2015. 29

work page 2015