A note on Parisian ruin under a hybrid observation scheme

Mohamed Amine Lkabous

arxiv: 1907.09993 · v1 · pith:A7XKQ5F6new · submitted 2019-07-23 · 🧮 math.PR · q-fin.RM

A note on Parisian ruin under a hybrid observation scheme

Mohamed Amine Lkabous This is my paper

Pith reviewed 2026-05-24 16:53 UTC · model grok-4.3

classification 🧮 math.PR q-fin.RM

keywords Parisian ruinhybrid observation schemefluctuation identitiessecond-generation scale functionsruin probabilityLévy process

0 comments

The pith

Parisian ruin probabilities and fluctuation identities under hybrid observation are expressed using second-generation scale functions.

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

This paper examines Parisian ruin in a model where a process is observed at random Poisson times when positive and continuously when negative. It refines earlier calculations of ruin probabilities and derives additional identities for exit times and overshoots. These results are formulated entirely in terms of second-generation scale functions of the underlying process. A sympathetic reader would care because such identities allow explicit computation of ruin metrics without solving integro-differential equations directly.

Core claim

Under the hybrid observation scheme, where observation is Poissonian above zero and continuous below, the probability of Parisian ruin and other fluctuation quantities can be expressed in closed form using second-generation scale functions, improving upon the expressions in Li et al. (2016).

What carries the argument

Second-generation scale functions, which encode the fluctuation theory for the process and permit the identities to be written explicitly.

Load-bearing premise

The hybrid observation scheme (Poisson arrivals when healthy, continuous when below zero) and the definition of Parisian ruin (consecutive time below zero exceeding a fixed delay) are taken as given without further justification in the model setup.

What would settle it

A numerical simulation of paths under the hybrid scheme that produces ruin probabilities differing from those given by the second-generation scale function formulas would disprove the identities.

read the original abstract

In this paper, we study the concept of Parisian ruin under the hybrid observation scheme model introduced by Li et al. \cite{binetal2016}. Under this model, the process is observed at Poisson arrival times whenever the business is financially healthy and it is continuously observed when it goes below $0$. The Parisian ruin is then declared when the process stays below zero for a consecutive period of time greater than a fixed delay. We improve the result originally obtained in \cite{binetal2016} and we compute other fluctuation identities. All identities are given in terms of second-generation scale functions.

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 is a clean but narrow technical extension that improves the 2016 Parisian ruin result and adds a few more identities via second-generation scale functions.

read the letter

The main takeaway is that the paper takes the hybrid observation model from Li et al. 2016 and produces improved and additional fluctuation identities for the Parisian ruin probability and related quantities, all written explicitly in terms of second-generation scale functions. That is the actual new content: a handful of extra expressions beyond the one improved result mentioned in the abstract. The approach follows the standard scale-function route that is already established in fluctuation theory for Lévy processes, so the derivations sit on familiar ground. The paper does this part straightforwardly and without introducing new parameters or circular definitions. The identities are presented as direct consequences of the existing machinery, which keeps the contribution focused and checkable against the prior literature. The soft spots are proportionate to the scope. The hybrid scheme itself and the definition of Parisian ruin with a fixed delay are adopted without fresh justification, which is fine for a note but means the work does not address why this monitoring setup is preferable or how it generalizes. There is no broader application or comparison to other ruin models, so the advance stays inside the same narrow subfield. No internal contradictions appear in the stated claims, and the citation pattern is appropriate, pointing back to the 2016 paper and the standard scale-function references. This is really only useful to specialists already working on insurance risk models that combine Poisson and continuous observation or that use delayed ruin definitions. A reader outside that niche will not find new concepts or methods. It deserves a serious referee because the claims are concrete, the method is reproducible in principle, and the incremental identities can be verified against the existing theory even if the overall impact remains modest.

Referee Report

0 major / 3 minor

Summary. The manuscript studies Parisian ruin under the hybrid observation scheme of Li et al. (2016), in which a Lévy process is monitored at Poisson arrival times while positive and continuously while negative. Parisian ruin occurs when the process remains below zero for a consecutive time exceeding a fixed delay. The authors improve the earlier result by deriving additional fluctuation identities, all expressed in terms of second-generation scale functions.

Significance. If the derivations are correct, the work supplies a more complete collection of explicit identities for ruin-related quantities in the hybrid-monitoring model. Because the expressions remain within the standard scale-function framework of fluctuation theory, they preserve the possibility of numerical evaluation and further analytic extensions without introducing new parameters or ad-hoc assumptions.

minor comments (3)

[Introduction] The abstract states that the identities improve upon Li et al. (2016), but the introduction does not quantify the improvement (e.g., which earlier identity is recovered as a special case or which new quantity is obtained). A short comparison paragraph would clarify the advance.
[Preliminaries] Notation for the second-generation scale functions is introduced without an explicit reference to the defining integral or Laplace-transform relation; adding the defining equation (even if standard) would aid readers who are not specialists in the sub-field.
[Model setup] The hybrid observation scheme is described in words; an equation or diagram showing the intensity of the Poisson observation process as a function of the current level would remove any ambiguity about the transition at zero.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained in fluctuation theory

full rationale

The paper derives fluctuation identities for Parisian ruin under the hybrid observation scheme by expressing them in terms of second-generation scale functions, which are standard pre-existing objects from Lévy process fluctuation theory (as referenced from prior literature such as Li et al. 2016). No step reduces a claimed prediction or identity to a fitted parameter or self-defined quantity by construction; the central contribution is an incremental mathematical extension without load-bearing self-citations, ansatz smuggling, or renaming of known results. The derivation chain remains independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard fluctuation theory for spectrally negative Lévy processes; no new free parameters, axioms, or invented entities are introduced in the abstract.

axioms (1)

standard math Standard properties of scale functions for spectrally negative Lévy processes hold and can be used to express ruin identities.
The abstract states all identities are given in terms of second-generation scale functions, invoking the established theory without re-deriving it.

pith-pipeline@v0.9.0 · 5616 in / 1162 out tokens · 60328 ms · 2026-05-24T16:53:37.665985+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.

All identities are given in terms of second-generation scale functions... Θ(q)(x;r,λ) ... hybrid scale function
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Parisian ruin under a hybrid observation scheme... Λ(p)(x;r,s) = ∫ W(p,s)x(x+z) z/r P(Xr ∈ dz)

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

16 extracted references · 16 canonical work pages · 1 internal anchor

[1]

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. MR3474819

work page 2016
[2]

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
[3]

Bertoin, Lévy processes, Cambridge University Press, 1996

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

work page 1996
[4]

Czarna and J.-F

I. Czarna and J.-F. Renaud, A note on parisian ruin with an ultimate bankruptcy level for Lévy insurance risk processes, Statist. Probab. Lett. (2016)

work page 2016
[5]

On the distribution of cumulative Parisian ruin

H. Guérin and J.-F. Renaud, On the distribution of cumulative Parisian ruin , arXiv:1509.06857 [math.PR]

work page internal anchor Pith review Pith/arXiv arXiv
[6]

Landriault, B

D. Landriault, B. Li, J. T.Y. Wong, and D. Xu, Poissonian potential measures for Lévy risk models , Insurance: Mathematics and Economics 82 (2018), 152 –166

work page 2018
[7]

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
[8]

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

work page 2014
[9]

B. Li, G. E. Willmot, and J. T. Y. Wong, A temporal approach to the Parisian risk model , J. Appl. Probab. 55 (2018), no. 1, 302–317

work page 2018
[10]

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. MR3257360

work page 2014
[11]

M. A. Lkabous, I. Czarna, and J.-F. Renaud, Parisian ruin for a refracted Lévy process , Insurance Math. Econom. 74 (2017), 153–163. MR3648884

work page 2017
[12]

M. A. Lkabous and J.-F. Renaud, A VaR-type risk measure derived from cumulative Parisian ru in for the classical risk model , Risks 6 (2018), no. 3

work page 2018
[13]

, A uniﬁed approach to ruin probabilities with delay for spect rally negative Lévy processes (submit- ted)

work page
[14]

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
[15]

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
[16]

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. Dépar tement de ma théma tiques, Université du Québec à Montréal (UQAM), 201 a v. Président- Kennedy, Montréal (Québec) H2X 3Y7, Canada E-mail address : lkabous.moh...

work page 2014

[1] [1]

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. MR3474819

work page 2016

[2] [2]

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

[3] [3]

Bertoin, Lévy processes, Cambridge University Press, 1996

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

work page 1996

[4] [4]

Czarna and J.-F

I. Czarna and J.-F. Renaud, A note on parisian ruin with an ultimate bankruptcy level for Lévy insurance risk processes, Statist. Probab. Lett. (2016)

work page 2016

[5] [5]

On the distribution of cumulative Parisian ruin

H. Guérin and J.-F. Renaud, On the distribution of cumulative Parisian ruin , arXiv:1509.06857 [math.PR]

work page internal anchor Pith review Pith/arXiv arXiv

[6] [6]

Landriault, B

D. Landriault, B. Li, J. T.Y. Wong, and D. Xu, Poissonian potential measures for Lévy risk models , Insurance: Mathematics and Economics 82 (2018), 152 –166

work page 2018

[7] [7]

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

[8] [8]

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

work page 2014

[9] [9]

B. Li, G. E. Willmot, and J. T. Y. Wong, A temporal approach to the Parisian risk model , J. Appl. Probab. 55 (2018), no. 1, 302–317

work page 2018

[10] [10]

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. MR3257360

work page 2014

[11] [11]

M. A. Lkabous, I. Czarna, and J.-F. Renaud, Parisian ruin for a refracted Lévy process , Insurance Math. Econom. 74 (2017), 153–163. MR3648884

work page 2017

[12] [12]

M. A. Lkabous and J.-F. Renaud, A VaR-type risk measure derived from cumulative Parisian ru in for the classical risk model , Risks 6 (2018), no. 3

work page 2018

[13] [13]

, A uniﬁed approach to ruin probabilities with delay for spect rally negative Lévy processes (submit- ted)

work page

[14] [14]

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

[15] [15]

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

[16] [16]

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. Dépar tement de ma théma tiques, Université du Québec à Montréal (UQAM), 201 a v. Président- Kennedy, Montréal (Québec) H2X 3Y7, Canada E-mail address : lkabous.moh...

work page 2014