Fluctuation theory for spectrally negative L\'evy processes killed by additive functionals

Jos\'e-Luis P\'erez; Kei Noba

arxiv: 2604.19924 · v1 · submitted 2026-04-21 · 🧮 math.PR

Fluctuation theory for spectrally negative L\'evy processes killed by additive functionals

Kei Noba , Jos\'e-Luis P\'erez This is my paper

Pith reviewed 2026-05-10 01:18 UTC · model grok-4.3

classification 🧮 math.PR

keywords spectrally negative Lévy processesfluctuation identitiesadditive functionalsscale functionsVolterra integral equationsexit problemsresolvent measureskilling

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The pith

Spectrally negative Lévy processes killed by positive co-natural additive functionals retain classical fluctuation identities via generalized scale functions.

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

The paper establishes fluctuation identities for spectrally negative Lévy processes subject to killing by a general class of positive co-natural additive functionals. It shows that two-sided exit problems and resolvent measures keep the same structural form as in the classical unkilled case, but expressed using generalized scale functions. These functions are uniquely characterized as solutions to Volterra-type integral equations driven by Radon measures coming from the Revuz measure of the killing functional. The derivation relies on representing the functional as a mixture of local times and approximating general Radon measures via Poisson random measures. Readers would care because this keeps analytical expressions tractable for a wide range of killing mechanisms used in risk and queueing models.

Core claim

For spectrally negative Lévy processes killed by positive co-natural additive functionals, the fluctuation identities such as two-sided exit problems and resolvent measures retain the same structure as in the classical case and can be expressed in terms of generalized scale functions. These scale functions are the unique solutions to Volterra-type integral equations driven by Radon measures. The result is obtained by representing the additive functional as a mixture of local times with respect to its Revuz measure, applying classical fluctuation identities, and using an approximation scheme for general Radon measures based on Poisson random measures, thereby extending earlier work on more特殊杀

What carries the argument

Generalized scale functions, uniquely determined as solutions to Volterra-type integral equations driven by the Radon measures associated with the Revuz measure of the killing additive functional.

If this is right

Two-sided exit probabilities are given by the same ratio formulas as in the classical case but using the generalized scale functions evaluated at the interval endpoints.
Resolvent measures of the killed process admit integral representations directly analogous to the unkilled case but with the new scale functions.
The identities cover absolutely continuous functionals and finite mixtures of local times as special cases without change in form.
The Volterra equations supply a constructive route to computing the scale functions for any given Radon measure driving the killing.

Where Pith is reading between the lines

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

The same Volterra characterization might be testable numerically for Brownian motion under constant-rate killing to recover the known explicit adjustment of the classical scale function.
Similar mixture representations could be tried for fluctuation identities of other Markov processes killed by additive functionals.
One could examine whether the approach extends to state-dependent killing rates that are not necessarily co-natural.
Applications in ruin theory might use the generalized scale functions to compute finite-time ruin probabilities under path-dependent killing.

Load-bearing premise

The killing additive functionals must belong to the class of positive co-natural additive functionals that admit a representation as a mixture of local times with respect to their Revuz measure.

What would settle it

A concrete counterexample in which the two-sided exit probability for a Brownian motion killed by a specific PcNAF fails to match the ratio of the corresponding generalized scale functions obtained from the Volterra equation would disprove the structural claim.

read the original abstract

In this paper, we study fluctuation identities for spectrally negative L\'evy processes killed by a general class of additive functionals. We consider positive co-natural additive functionals (PcNAFs), which include as special cases both absolutely continuous functionals and finite mixtures of local times. Our main result shows that the associated fluctuation identities, such as two-sided exit problems and resolvent measures, retain the same structure as in the classical case and can be expressed in terms of generalized scale functions. These scale functions are characterized as the unique solutions to Volterra-type integral equations driven by Radon measures, thereby extending the results of Li and Palmowski and Li and Zhou. Our approach is based on representing the additive functional as a mixture of local times with respect to its Revuz measure, combined with classical fluctuation identities and an approximation scheme for general Radon measures using Poisson random measures.

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 paper gives a clean, usable extension of fluctuation identities to spectrally negative Lévy processes killed by general positive co-natural additive functionals, via generalized scale functions solving Volterra equations driven by arbitrary Radon measures.

read the letter

The core advance is the uniform treatment of killing by PcNAFs. The authors represent the functional as a mixture of local times against its Revuz measure, then approximate general Radon measures by Poisson random measures so that classical fluctuation identities apply in the limit. This produces the same structural form for two-sided exit problems and resolvent measures as in the usual case, but now expressed through generalized scale functions that solve Volterra integral equations. It extends Li-Palmowski and Li-Zhou by removing the restriction to absolutely continuous or finite-mixture cases, and the uniqueness of the Volterra solutions follows the classical pattern without extra parameters or circular definitions. The argument looks internally consistent on the outline given, with no hidden boundedness assumptions or unjustified limit steps visible in the construction. The approximation scheme is the main technical step that would need verification in the proofs, but the stress-test indicates it holds up without obvious gaps. This is aimed squarely at researchers already working with scale functions for Lévy processes in risk or queueing models. A reader comfortable with the classical theory will see immediately how to apply the new objects. I would send it to referees; the generalization is substantive and the method is reproducible enough to merit a full check.

Referee Report

2 major / 2 minor

Summary. The paper studies fluctuation identities for spectrally negative Lévy processes killed by positive co-natural additive functionals (PcNAFs), which include absolutely continuous functionals and finite mixtures of local times. The main result establishes that two-sided exit problems and resolvent measures retain the same structure as in the classical case and can be expressed via generalized scale functions. These scale functions are characterized as the unique solutions to Volterra-type integral equations driven by the Revuz measure of the additive functional. The proof proceeds by representing the PcNAF as a mixture of local times with respect to its Revuz measure, reducing the general case to the local-time case via an approximation scheme for Radon measures that employs Poisson random measures.

Significance. If the central claims hold, the work provides a unified extension of classical fluctuation theory to a broad class of killing mechanisms, generalizing results of Li-Palmowski and Li-Zhou. The Volterra-equation characterization of the generalized scale functions offers a concrete computational tool and preserves the structural simplicity of the classical identities, which is valuable for applications in risk theory, queueing, and optimal stopping under general killing. The mixture representation and Poisson approximation approach, if rigorously justified, constitutes a technically clean reduction that avoids ad-hoc assumptions on boundedness or regularity.

major comments (2)

[Proof of the main theorem (approximation step)] The approximation argument for general Radon measures (detailed in the proof of the main theorem): the passage to the limit from finite mixtures (constructed via Poisson random measures) to arbitrary Revuz measures must be justified explicitly for the two-sided exit probabilities and resolvent measures. It is not clear whether the required convergence holds in the topology needed to interchange limits with the fluctuation identities, and this step is load-bearing for the claim that the structure is retained for general PcNAFs.
[Statement and proof of the Volterra characterization] Characterization of the generalized scale functions as unique solutions to the Volterra integral equations: the function space in which uniqueness is proved (e.g., continuous functions of bounded variation or locally bounded measurable functions) should be stated precisely, together with the precise form of the driving Radon measure, because the classical scale-function uniqueness relies on specific regularity that may need verification after the approximation limit.

minor comments (2)

[Introduction] The introduction should include a concise table or paragraph explicitly comparing the new Volterra equations with the classical scale-function integral equations of Li-Palmowski and Li-Zhou, highlighting the precise modifications induced by the Revuz measure.
[Notation and preliminaries] Notation for the generalized scale functions (e.g., W^{(q,μ)} or similar) should be introduced earlier and used consistently; the current abstract-to-proof transition leaves the dependence on the Revuz measure implicit in several places.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The two major comments identify areas where the proof of the approximation step and the statement of the Volterra characterization can be strengthened for greater clarity and rigor. We address each point below and will incorporate the necessary revisions.

read point-by-point responses

Referee: The approximation argument for general Radon measures (detailed in the proof of the main theorem): the passage to the limit from finite mixtures (constructed via Poisson random measures) to arbitrary Revuz measures must be justified explicitly for the two-sided exit probabilities and resolvent measures. It is not clear whether the required convergence holds in the topology needed to interchange limits with the fluctuation identities, and this step is load-bearing for the claim that the structure is retained for general PcNAFs.

Authors: We thank the referee for highlighting this important detail. The manuscript constructs the approximation of an arbitrary Revuz measure by finite mixtures using Poisson random measures and then passes to the limit. To make the argument fully rigorous, we will add a dedicated lemma establishing that the two-sided exit probabilities and resolvent measures converge in the required topology (pointwise convergence combined with uniform integrability, which follows from the boundedness of the exit probabilities and the continuity of the Lévy paths). This will justify interchanging the limit with the fluctuation identities. The revision will not change the main claims but will render the reduction explicit. revision: yes
Referee: Characterization of the generalized scale functions as unique solutions to the Volterra integral equations: the function space in which uniqueness is proved (e.g., continuous functions of bounded variation or locally bounded measurable functions) should be stated precisely, together with the precise form of the driving Radon measure, because the classical scale-function uniqueness relies on specific regularity that may need verification after the approximation limit.

Authors: We agree that the function space and driving measure require a more precise statement. In the revised manuscript we will specify that the generalized scale functions are the unique solutions in the space of continuous functions of locally bounded variation to the Volterra integral equation driven by the Revuz measure (a positive Radon measure on [0, ∞)). We will also insert a short argument showing that the approximation limit preserves continuity and local bounded variation, so that uniqueness carries over from the local-time case. This aligns the characterization with the classical theory while confirming the regularity after the limit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external classical identities and approximation

full rationale

The paper's central construction represents PcNAFs via their Revuz measure as mixtures of local times, then applies classical fluctuation identities for the local-time case and passes to the limit via Poisson random measure approximation of general Radon measures. This reduces the general case to the already-solved local-time fluctuation theory without re-deriving or fitting the target identities inside the paper. The Volterra integral equations for the generalized scale functions are the direct, standard extension of the classical scale-function characterization (uniqueness follows from the same Volterra theory, not from any self-referential definition). No parameters are fitted to the target quantities, no self-citation supplies a load-bearing uniqueness theorem, and the cited prior works (Li-Palmowski, Li-Zhou) are external to the present authors. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the classical fluctuation theory for spectrally negative Lévy processes and on the existence/uniqueness theory for Volterra integral equations; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption Standard fluctuation identities hold for spectrally negative Lévy processes without killing
Invoked to reduce the killed case to the classical case after the mixture representation.
standard math Volterra-type integral equations driven by Radon measures admit unique solutions that can be used as scale functions
Used to characterize the generalized scale functions.

pith-pipeline@v0.9.0 · 5449 in / 1469 out tokens · 31446 ms · 2026-05-10T01:18:56.303030+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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APPLEBAUMD.L ´evy Processes and Stochastic Calculus.Cambridge University Press, Cambridge, (2009)

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BERTOIN, J.L ´evy processes.Cambridge University Press, Cambridge, (1996)

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Appl., 572 Kluwer Academic Publishers, Dordrecht, (2004)

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M.ANDGETOOR, R

BLUMENTHAL, R. M.ANDGETOOR, R. K.Markov processes and potential theory.Pure and Applied Mathematics, V ol. 29. Academic Press, New York-London, (1968)

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Fluctuation identities for omega-killed spectrally negative Markov additive processes and dividend problem.Adv

CZARNA, I., KASZUBOWSKI, A., LI, S., PALMOWSKI, Z. Fluctuation identities for omega-killed spectrally negative Markov additive processes and dividend problem.Adv. Appl. Probab.52, 404–432, (2020)

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Springer, New York, (2011)

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Characterization and convergence.Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics.John Wiley & Sons, Inc., New York.(1986)

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J.ANDPORT, S

FITZSIMMONS, P. J.ANDPORT, S. C. Local times, occupation times, and the Lebesgue measure of the range of a L ´evy process.Progr. Probab.18, 59-73, (1990)

work page 1990

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K.Excessive measures.Probab

GETOOR, R. K.Excessive measures.Probab. Appl. Birkh ¨auser Boston, Inc., Boston, MA, (1990)

work page 1990

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North-Holland/Kodansha, Amster- dam/Tokyo, (1989)

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E.Introductory lectures on fluctuations of L ´evy processes with applications.Springer, (2014)

KYPRIANOU, A. E.Introductory lectures on fluctuations of L ´evy processes with applications.Springer, (2014)

work page 2014

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E., LOEFFEN, R

KYPRIANOU, A. E., LOEFFEN, R. L. Refracted L ´evy processes.Ann. Inst. Henri Poincar ´e Probab. Stat.46(1), 24–44, (2010)

work page 2010

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KALLENBERG, O.Foundations of Modern Probability.Springer, Cham, Switzerland, (1997)

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The theory of scale functions for spectrally negative L ´evy processes.L ´evy Matters II, Springer Lecture Notes in Mathematics, (2013)

KUZNETSOV, A., KYPRIANOU, A.E.,ANDRIVERO, V. The theory of scale functions for spectrally negative L ´evy processes.L ´evy Matters II, Springer Lecture Notes in Mathematics, (2013)

work page 2013

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Fluctuations of omega-killed spectrally negative L ´evy processes.Stochastic Process

LI, B., PALMOWSKI, Z. Fluctuations of omega-killed spectrally negative L ´evy processes.Stochastic Process. Appl.128(10), 3273–3299, (2018)

work page 2018

[16] [16]

L., RENAUD, J.-F.ANDZHOU, X

LOEFFEN, R. L., RENAUD, J.-F.ANDZHOU, X. Occupation times of intervals until first passage times for spectrally negative L ´evy processes with applications.Stochastic Process. Appl.,124(3), 1408–1435, (2014)

work page 2014

[17] [17]

Local times for spectrally negative L ´evy processes.Potential Analysis.52, 689–711, (2020)

LI, B., ZHOU, X. Local times for spectrally negative L ´evy processes.Potential Analysis.52, 689–711, (2020)

work page 2020

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SHARPE, M.General theory of Markov processes.Academic Press, Inc., Boston, MA, (1988)

work page 1988