Poissonian occupation times of spectrally negative L\'evy processes with applications

Mohamed Amine Lkabous

arxiv: 1907.09990 · v1 · pith:JHCUCEAKnew · submitted 2019-07-23 · 🧮 math.PR · q-fin.RM

Poissonian occupation times of spectrally negative L\'evy processes with applications

Mohamed Amine Lkabous This is my paper

Pith reviewed 2026-05-24 16:56 UTC · model grok-4.3

classification 🧮 math.PR q-fin.RM

keywords Poissonian occupation timesspectrally negative Lévy processesParisian implementation delaysinsurance risk modelsoccupation timesfluctuation theoryruin probabilities

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The pith

Poissonian occupation times for spectrally negative Lévy processes are accumulated only at independent Poisson arrival epochs and correspond to ruin quantities in insurance models with Parisian delays.

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

The paper defines a version of occupation time below zero in which a spectrally negative Lévy process is checked for negativity only at the random times of an independent Poisson process. This discrete sampling produces quantities that recover the classical continuously observed occupation times as special cases. The central application establishes an exact correspondence between these sampled times and the ruin indicators that arise when an insurance risk process incorporates implementation delays of the Parisian type.

Core claim

Poissonian occupation times below level zero are introduced for spectrally negative Lévy processes by accumulating time only when the process is found negative at the arrival epochs of an independent Poisson process; the resulting objects satisfy fluctuation identities that extend the continuously observed case and map directly onto the distribution of ruin times in risk models that implement Parisian delays.

What carries the argument

Poissonian occupation time below zero, formed by integrating the indicator that the Lévy process is negative at the points of an independent Poisson process.

If this is right

The Poissonian construction yields new identities for occupation times that reduce to known continuous-observation formulas when the Poisson intensity tends to infinity.
Joint distributions involving the Poissonian occupation time, the position at an exponential time, and the overshoot are obtained in explicit form.
The correspondence supplies Laplace transforms for the time to Parisian ruin and related quantities in insurance models driven by spectrally negative Lévy processes.
The approach extends existing fluctuation-theory results on exit problems and occupation times to a Poisson-sampled observation regime.

Where Pith is reading between the lines

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

The same Poisson sampling device could be applied to other additive functionals such as the integral of the process or the local time at zero.
Numerical path simulation under Poissonian monitoring may offer computational advantages over continuous monitoring for certain Lévy models.
The link to delayed ruin suggests that similar embedding arguments could treat more general forms of observation delay in applied probability.

Load-bearing premise

The Poisson process that samples the sign of the Lévy path must be independent of the driving process and must have a constant rate.

What would settle it

An explicit calculation, for Brownian motion with drift, showing that the Laplace transform of the Poissonian occupation time differs from the claimed expression when the Poisson rate is made state-dependent.

Figures

Figures reproduced from arXiv: 1907.09990 by Mohamed Amine Lkabous.

**Figure 1.** Figure 1: Illustration of Poissonian occupation time below 0. 2. Preliminaries In this section, we present the necessary background material on spectrally negative Lévy processes. 2.1. Lévy insurance risk processes. A Lévy insurance risk process X is a process with stationary and independent increments and no positive jumps. To avoid trivialities, we exclude the case where X has monotone paths. As the Lévy process X… view at source ↗

read the original abstract

In this paper, we introduce the concept of \emph{Poissonian occupation times} below level $0$ of spectrally negative L\'evy processes. In this case, occupation time is accumulated only when the process is observed to be negative at arrival epochs of an independent Poisson process. Our results extend some well known continuously observed quantities involving occupation times of spectrally negative L\'evy processes. As an application, we establish a link between Poissonian occupation times and insurance risk models with Parisian implementation delays.

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

The paper defines Poissonian occupation times via independent Poisson sampling of a spectrally negative Lévy process and links the object to Parisian-delay insurance models.

read the letter

The central new item is the Poissonian occupation time below zero, accumulated only at the arrival times of an exogenous Poisson process. This is distinct from the classical continuous occupation time, though it converges to it as the Poisson rate grows. The paper applies standard fluctuation identities for spectrally negative Lévy processes to this sampled version and states an application to risk models with implementation delays modeled by the Poisson points. That modeling choice is stated plainly: constant rate, independent of the driving process. The derivations appear to follow directly from the existing literature on exit times and resolvents, so the technical lift looks modest but the object itself is new. The link to Parisian delays is the main applied payoff and seems a reasonable fit for the sampling interpretation. The abstract gives no indication of circularity or fitted parameters; the quantities are defined from the process paths and the Poisson arrivals. One limitation is that the constant-rate independence is required for the identities to hold in closed form, which narrows the modeling scope but is not hidden. Without the full proofs it is impossible to confirm there are no gaps in the extension steps, but nothing in the stated claims contradicts the cited continuous-time results. The work is aimed at readers already using Lévy fluctuation theory in insurance or queueing; those readers will see a usable new quantity and a concrete application. It is worth sending to referees because the definition is clean, the extension is explicit, and the application claim can be checked against the derivations.

Referee Report

0 major / 2 minor

Summary. The paper introduces Poissonian occupation times below level 0 for spectrally negative Lévy processes, accumulated only at arrival epochs of an independent Poisson process when the process is negative. It extends known fluctuation identities for continuous occupation times and establishes a link to insurance risk models with Parisian implementation delays.

Significance. If the derivations are correct, the work extends the fluctuation theory toolkit for spectrally negative Lévy processes to a discrete Poissonian sampling regime. This has direct modeling value in risk theory where continuous monitoring is replaced by random observations, and the explicit link to Parisian-delay ruin models is a concrete applied contribution. The modeling choice of an exogenous independent Poisson process yields identities that parallel the continuous case without introducing fitted parameters.

minor comments (2)

[Abstract] Abstract: the phrase 'extend some well known continuously observed quantities' would benefit from a parenthetical reference to the specific classical identities being generalized (e.g., the occupation-time formula of Kyprianou or Ivanovs).
The independence and constant-rate assumptions on the Poisson process are correctly presented as modeling prerequisites rather than derived results; a brief remark on robustness under mild dependence would strengthen the applied section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript. The summary accurately reflects the paper's contributions on Poissonian occupation times and the link to Parisian-delay risk models. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines Poissonian occupation times directly via an exogenous independent Poisson process sampling the sign of a spectrally negative Lévy process. It extends standard occupation-time identities for Lévy processes and applies the construction to Parisian-delay risk models. No equation reduces a claimed prediction or new quantity to a fitted parameter, self-citation, or ansatz imported from the authors' prior work. The independence and constant-rate assumptions are stated as modeling prerequisites, not derived results. The central objects and identities remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard fluctuation theory of spectrally negative Lévy processes and the independence of an auxiliary Poisson process; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The driving process is spectrally negative (no positive jumps).
Invoked to guarantee the existence of scale functions used in the occupation-time identities.
domain assumption The Poisson sampling process is independent of the Lévy process and has constant intensity.
Required for the Poissonian occupation time to be a well-defined Markovian functional.

pith-pipeline@v0.9.0 · 5605 in / 1247 out tokens · 18671 ms · 2026-05-24T16:56:12.473627+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.

We define the Poissonian occupation time below 0 … OX_{t,λ} = ∑ (τ₀⁺ ∘ θ_{T_n}) 1_{X_{T_n}<0, T_n<t} where T_n are arrival times of an independent Poisson process with intensity λ>0
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

lim λ→∞ recovers the continuously observed occupation time identities of Landriault et al. [12] and Loeffen et al. [20]

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

21 extracted references · 21 canonical work pages

[1]

Albrecher and J

H. Albrecher and J. Ivanovs, Strikingly simple identities relating exit problems for Lé vy processes under continuous and Poisson observations , Stochastic Process. Appl. 127 (2017), no. 2, 643–656

work page 2017
[2]

Albrecher, J

H. Albrecher, J. Ivanovs, and X. Zhou, Exit identities for Lévy processes observed at Poisson arri val times , Bernoulli 22 (2016), no. 3, 1364–1382

work page 2016
[3]

E. J. Baurdoux, J. C. Pardo, J. L. Pérez, and J.-F. Renaud, Gerber-Shiu distribution at Parisian ruin for Lévy insurance risk processes , J. Appl. Probab. (2016)

work page 2016
[4]

Bertoin, Lévy processes, Cambridge University Press, 1996

J. Bertoin, Lévy processes, Cambridge University Press, 1996

work page 1996
[5]

Bladt, B

M. Bladt, B. F. Nielsen, and O. Peralta, Parisian types of ruin probabilities for a class of depen- dent risk-reserve processes , Scandinavian Actuarial Journal 2019 (2019), no. 1, 32–61, available at https://doi.org/10.1080/03461238.2018.1483420

work page doi:10.1080/03461238.2018.1483420 2019
[6]

Frostig and A

E. Frostig and A. Keren-Pinhasik, Parisian ruin with erlang delay and a lower bankruptcy barri er, Method- ology and Computing in Applied Probability (2019Jan)

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Guérin and J.-F

H. Guérin and J.-F. Renaud, Joint distribution of a spectrally negative Lévy process an d its occupation time, with step option pricing in view , Adv. in Appl. Probab. (2016)

work page 2016
[8]

, On the distribution of cumulative Parisian ruin , Insurance Math. Econom. 73 (2017), 116–123

work page 2017
[9]

Kuznetsov, A

A. Kuznetsov, A. E. Kyprianou, and V. Rivero, The theory of scale functions for spectrally negative Lévy processes, Lévy Matters - Springer Lecture Notes in Mathematics, 2012

work page 2012
[10]

A. E. Kyprianou, Fluctuations of Lévy processes with applications - Introdu ctory lectures, Second, Univer- sitext, Springer, Heidelberg, 2014

work page 2014
[11]

Landriault, B

D. Landriault, B. Li, and M.A. Lkabous, On occupation times in the red of Lévy risk models (submitted)

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[12]

Landriault, J.-F

D. Landriault, J.-F. Renaud, and X. Zhou, Occupation times of spectrally negative Lévy processes wit h applications, Stochastic Process. Appl. 121 (2011), no. 11, 2629–2641

work page 2011
[13]

, An insurance risk model with Parisian implementation delay s, Methodol. Comput. Appl. Probab. 16 (2014), no. 3, 583–607

work page 2014
[14]

Li and Z

B. Li and Z. Palmowski, Fluctuations of omega-killed spectrally negative Lévy pro cesses, Stochastic Pro- cesses and their Applications (2017)

work page 2017
[15]

Li and X

Y. Li and X. Zhou, On pre-exit joint occupation times for spectrally negative Lévy processes, Statist. Probab. Lett. 94 (2014), 48–55

work page 2014
[16]

Y. Li, X. Zhou, and N. Zhu, Two-sided discounted potential measures for spectrally negative Lévy processes, Statist. Probab. Lett. 100 (2015), 67–76

work page 2015
[17]

M. A. Lkabous, A note on Parisian ruin under a hybrid observation scheme , Statistics & Probability Letters 145 (2019), 147 –157

work page 2019
[18]

R. L. Loeﬀen, I. Czarna, and Z. Palmowski, Parisian ruin probability for spectrally negative Lévy pro cesses, Bernoulli 19 (2013), no. 2, 599–609

work page 2013
[19]

R. L. Loeﬀen, Z. Palmowski, and B. A. Surya, Discounted penalty function at Parisian ruin for Lévy insurance risk process, Insurance Math. Econom. (2017)

work page 2017
[20]

R. L. Loeﬀen, J.-F. Renaud, and X. Zhou, Occupation times of intervals until ﬁrst passage times for spectrally negative Lévy processes , Stochastic Process. Appl. 124 (2014), no. 3, 1408–1435

work page 2014
[21]

B. A. Surya, Evaluating scale functions of spectrally negative lévy pro cesses, Journal of Applied Probability 45 (2008), no. 1, 135–149. 18 Depar tment of Sta tistics and Actuarial Science, Universit y of W a terloo, W a terloo, ON, N2L 3G1, Canada E-mail address : mohamed.amine.lkabous@uwaterloo.ca 19

work page 2008

[1] [1]

Albrecher and J

H. Albrecher and J. Ivanovs, Strikingly simple identities relating exit problems for Lé vy processes under continuous and Poisson observations , Stochastic Process. Appl. 127 (2017), no. 2, 643–656

work page 2017

[2] [2]

Albrecher, J

H. Albrecher, J. Ivanovs, and X. Zhou, Exit identities for Lévy processes observed at Poisson arri val times , Bernoulli 22 (2016), no. 3, 1364–1382

work page 2016

[3] [3]

E. J. Baurdoux, J. C. Pardo, J. L. Pérez, and J.-F. Renaud, Gerber-Shiu distribution at Parisian ruin for Lévy insurance risk processes , J. Appl. Probab. (2016)

work page 2016

[4] [4]

Bertoin, Lévy processes, Cambridge University Press, 1996

J. Bertoin, Lévy processes, Cambridge University Press, 1996

work page 1996

[5] [5]

Bladt, B

M. Bladt, B. F. Nielsen, and O. Peralta, Parisian types of ruin probabilities for a class of depen- dent risk-reserve processes , Scandinavian Actuarial Journal 2019 (2019), no. 1, 32–61, available at https://doi.org/10.1080/03461238.2018.1483420

work page doi:10.1080/03461238.2018.1483420 2019

[6] [6]

Frostig and A

E. Frostig and A. Keren-Pinhasik, Parisian ruin with erlang delay and a lower bankruptcy barri er, Method- ology and Computing in Applied Probability (2019Jan)

work page

[7] [7]

Guérin and J.-F

H. Guérin and J.-F. Renaud, Joint distribution of a spectrally negative Lévy process an d its occupation time, with step option pricing in view , Adv. in Appl. Probab. (2016)

work page 2016

[8] [8]

, On the distribution of cumulative Parisian ruin , Insurance Math. Econom. 73 (2017), 116–123

work page 2017

[9] [9]

Kuznetsov, A

A. Kuznetsov, A. E. Kyprianou, and V. Rivero, The theory of scale functions for spectrally negative Lévy processes, Lévy Matters - Springer Lecture Notes in Mathematics, 2012

work page 2012

[10] [10]

A. E. Kyprianou, Fluctuations of Lévy processes with applications - Introdu ctory lectures, Second, Univer- sitext, Springer, Heidelberg, 2014

work page 2014

[11] [11]

Landriault, B

D. Landriault, B. Li, and M.A. Lkabous, On occupation times in the red of Lévy risk models (submitted)

work page

[12] [12]

Landriault, J.-F

D. Landriault, J.-F. Renaud, and X. Zhou, Occupation times of spectrally negative Lévy processes wit h applications, Stochastic Process. Appl. 121 (2011), no. 11, 2629–2641

work page 2011

[13] [13]

, An insurance risk model with Parisian implementation delay s, Methodol. Comput. Appl. Probab. 16 (2014), no. 3, 583–607

work page 2014

[14] [14]

Li and Z

B. Li and Z. Palmowski, Fluctuations of omega-killed spectrally negative Lévy pro cesses, Stochastic Pro- cesses and their Applications (2017)

work page 2017

[15] [15]

Li and X

Y. Li and X. Zhou, On pre-exit joint occupation times for spectrally negative Lévy processes, Statist. Probab. Lett. 94 (2014), 48–55

work page 2014

[16] [16]

Y. Li, X. Zhou, and N. Zhu, Two-sided discounted potential measures for spectrally negative Lévy processes, Statist. Probab. Lett. 100 (2015), 67–76

work page 2015

[17] [17]

M. A. Lkabous, A note on Parisian ruin under a hybrid observation scheme , Statistics & Probability Letters 145 (2019), 147 –157

work page 2019

[18] [18]

R. L. Loeﬀen, I. Czarna, and Z. Palmowski, Parisian ruin probability for spectrally negative Lévy pro cesses, Bernoulli 19 (2013), no. 2, 599–609

work page 2013

[19] [19]

R. L. Loeﬀen, Z. Palmowski, and B. A. Surya, Discounted penalty function at Parisian ruin for Lévy insurance risk process, Insurance Math. Econom. (2017)

work page 2017

[20] [20]

R. L. Loeﬀen, J.-F. Renaud, and X. Zhou, Occupation times of intervals until ﬁrst passage times for spectrally negative Lévy processes , Stochastic Process. Appl. 124 (2014), no. 3, 1408–1435

work page 2014

[21] [21]

B. A. Surya, Evaluating scale functions of spectrally negative lévy pro cesses, Journal of Applied Probability 45 (2008), no. 1, 135–149. 18 Depar tment of Sta tistics and Actuarial Science, Universit y of W a terloo, W a terloo, ON, N2L 3G1, Canada E-mail address : mohamed.amine.lkabous@uwaterloo.ca 19

work page 2008