Ruin Probabilities for a Sparre Andersen Model with Investments: the Case of Annuity Payments

Platon Promyslov; Yuri Kabanov

arxiv: 2301.01966 · v1 · submitted 2023-01-05 · 🧮 math.PR

Ruin Probabilities for a Sparre Andersen Model with Investments: the Case of Annuity Payments

Yuri Kabanov , Platon Promyslov This is my paper

Pith reviewed 2026-05-24 09:54 UTC · model grok-4.3

classification 🧮 math.PR

keywords ruin probabilitySparre Andersen modelannuity paymentstwo-sided jumpssemi-Markov processesgeometric Lévy processinsurance investmentsasymptotic analysis

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The pith

Ruin probability asymptotics extend to annuity payments and two-sided jumps in Sparre Andersen models with investments.

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

This paper extends an earlier asymptotic result for ruin probabilities in a Sparre Andersen non-life insurance model that includes investments in a risky asset whose price follows a geometric Lévy process. It applies semi-Markov process techniques to cover the cases of annuity payments and risk processes with jumps in both directions. A sympathetic reader would care because these changes make the model closer to actual insurance products that pay benefits continuously and face risks from both gains and losses.

Core claim

Using the techniques of semi-Markov processes we extend the result of the mentioned paper to the case of annuities and models with two-sided jumps.

What carries the argument

Semi-Markov process techniques applied to the risk process, which carry the asymptotic ruin probability result over to annuity structures and bidirectional jumps.

Load-bearing premise

The semi-Markov process techniques developed for the original Sparre Andersen model with one-sided jumps and non-annuity payments carry over directly to annuity structures and two-sided jumps without requiring additional assumptions or new technical conditions.

What would settle it

A concrete example of an annuity payment schedule or a two-sided jump distribution where the original asymptotic formula for ruin probability fails to hold under exactly the same model conditions as the prior paper.

read the original abstract

This note is a complement to the paper by Eberlein, Kabanov, and Schmidt on the asymptotic of the ruin probability in a Sparre Andersen non-life insurance model with investments a risky asset whose price follows a geometric L\'evy process. Using the techniques of semi-Markov processes we extend the result of the mentioned paper to the case of annuities and models with two-sided jumps.

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

Short note applies prior semi-Markov methods to extend ruin asymptotics to annuities and two-sided jumps, but offers no explicit checks on whether the original conditions still suffice.

read the letter

This note extends the ruin probability results from the Eberlein-Kabanov-Schmidt paper on the Sparre Andersen model with geometric Lévy investments to the cases of annuity payments and two-sided jumps. It does so by invoking the same semi-Markov process techniques used earlier. The contribution is narrow but fills the specific gaps the prior work left open, and the positioning as a complement is accurate. The writing is direct and stays within the established framework without claiming broader novelty. The main limitation is the lack of detail on how the extension is carried out. Two-sided jumps allow the process to cross levels upward, which can change the ladder-height structure and regeneration points that define the semi-Markov kernel in the one-sided case. Continuous annuity payments also alter the behavior between jumps. The abstract states the techniques carry over but supplies no new conditions, error bounds, or verification steps to confirm this. If the full text contains those checks and shows the asymptotics remain valid under the original net-profit criterion, the claim holds; otherwise the transfer is assumed rather than demonstrated. The paper is aimed at specialists already working on ruin probabilities in insurance models with investments. Such readers will see a useful incremental update. It is worth sending to peer review because the underlying model class is established and the extension is well-defined, even if the technical justification needs closer scrutiny in revision.

Referee Report

2 major / 1 minor

Summary. This short note claims to extend the asymptotic ruin probability results from Eberlein, Kabanov and Schmidt (on a Sparre Andersen model with investments driven by a geometric Lévy process) to the cases of continuous annuity payments and two-sided jumps, by applying semi-Markov process techniques.

Significance. If the extension is rigorously justified, the result would broaden the scope of the original asymptotics to insurance models with ongoing payment streams and bidirectional jumps, which are relevant for certain annuity and investment-risk settings.

major comments (2)

[Abstract / main text] The note states that semi-Markov techniques extend the prior result but supplies neither the modified semi-Markov kernel nor the adjusted integro-differential equation that accounts for continuous annuity outflows between jumps; without these, it is impossible to check whether the original net-profit condition and Lévy-measure assumptions remain sufficient.
[Abstract / main text] Two-sided jumps allow the risk process to cross levels upward, which can eliminate the one-sided ladder-height regeneration points used in the original semi-Markov embedding; the manuscript contains no verification that the embedding and asymptotic analysis survive this change.

minor comments (1)

The note is extremely brief and does not restate the key assumptions of the referenced Eberlein-Kabanov-Schmidt paper that are being carried over.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and the opportunity to clarify our short note. We address each major comment below and indicate where revisions will be made to improve the presentation.

read point-by-point responses

Referee: [Abstract / main text] The note states that semi-Markov techniques extend the prior result but supplies neither the modified semi-Markov kernel nor the adjusted integro-differential equation that accounts for continuous annuity outflows between jumps; without these, it is impossible to check whether the original net-profit condition and Lévy-measure assumptions remain sufficient.

Authors: We agree that the brevity of the note omits an explicit derivation of the modified semi-Markov kernel and the adjusted integro-differential equation. The continuous annuity is incorporated as a deterministic negative drift between jumps; the kernel is obtained by composing the geometric Lévy evolution with this drift over the random inter-jump time, and the net-profit condition is adjusted by subtracting the annuity rate from the drift term. The Lévy-measure assumptions carry over directly. We will add a short section (or appendix) supplying the kernel and confirming sufficiency of the conditions. revision: yes
Referee: [Abstract / main text] Two-sided jumps allow the risk process to cross levels upward, which can eliminate the one-sided ladder-height regeneration points used in the original semi-Markov embedding; the manuscript contains no verification that the embedding and asymptotic analysis survive this change.

Authors: The note claims the extension to two-sided jumps but indeed supplies no explicit verification that the semi-Markov embedding and asymptotics remain valid. Upward crossings are handled by redefining the regeneration points via the first passage times of the Lévy-driven process (accounting for both positive and negative jumps), preserving the Markov property at claim epochs. We acknowledge the lack of detail and will insert a concise argument in the revision showing that the original asymptotic analysis extends under the same net-profit and moment conditions. revision: yes

Circularity Check

1 steps flagged

Central extension claim rests on self-citation to overlapping-author prior work without new verification

specific steps

self citation load bearing [abstract]
"This note is a complement to the paper by Eberlein, Kabanov, and Schmidt on the asymptotic of the ruin probability in a Sparre Andersen non-life insurance model with investments a risky asset whose price follows a geometric Lévy process. Using the techniques of semi-Markov processes we extend the result of the mentioned paper to the case of annuities and models with two-sided jumps."

The load-bearing claim (extension to annuities and two-sided jumps) is justified solely by reference to the prior work whose author list overlaps with the present paper; the note supplies no separate derivation, adjusted integro-differential equation, or verification that the original technical conditions remain sufficient for the new cases.

full rationale

The paper is explicitly a short complement note whose sole contribution is the assertion that semi-Markov techniques from the cited Eberlein-Kabanov-Schmidt paper extend directly to annuities and two-sided jumps. No equations, new kernel conditions, or independent checks appear in the provided text; the claim therefore reduces to the self-citation. This matches the self_citation_load_bearing pattern at the abstract level and produces a moderate circularity score, while still leaving room for the prior paper's own proofs to be non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5586 in / 1045 out tokens · 26453 ms · 2026-05-24T09:54:36.085289+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/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Using the techniques of semi-Markov processes we extend the result of the mentioned paper to the case of annuities and models with two-sided jumps.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Theorem 3.1 … E[M^β]=1, E[M^β(ln M)^+]<∞, E[|Q|^β]<∞ … Y_∞ solves Y_∞ =^d Q + M Y_∞ … lim u^β ¯G(u) …

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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.
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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

17 extracted references · 17 canonical work pages

[1]

Albrecher, H., Constantinescu, C., Thomann, E.: Asymptotic results for renewal risk models with risky investments. Stoch. Proc. Appl. 122 , 3767--3789 (2012)

work page 2012
[2]

Eberlein, E., Kabanov, Yu., Schmidt, T.: Ruin probabilities for a Sparre Andersen model with investments. Stoch. Proc. Appl. 144 , 72--84 (2022)

work page 2022
[3]

Some mathematical models of risk theory

Frolova A.G. Some mathematical models of risk theory

work page
[4]

Goldie, C.M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 , 126--166 (1991)

work page 1991
[5]

Springer, Berlin (1990)

Grandell, I.: Aspects of Risk Theory. Springer, Berlin (1990)

work page 1990
[6]

Finance Stoch

Kabanov, Yu., Pergamenshchikov, S.: In the insurance business risky investments are dangerous: the case of negative risk sums. Finance Stoch. 20 , 355--379 (2016)

work page 2016
[7]

Finance Stoch

Kabanov, Yu., Pergamenshchikov, S.: Ruin probabilities for a L\'evy-driven generalised Ornstein--Uhlenbeck process. Finance Stoch. 24 , 39--69 (2020)

work page 2020
[8]

Kabanov, Yu., Pukhlyakov, N.: Ruin probabilities with investments: smoothness, IDE and ODE, asymptotic behavior. J. Appl. Probab. 59 , 556--570 (2020)

work page 2020
[9]

Ruin probability under random

red Kalashnikov V., Norberg R. Ruin probability under random

work page
[10]

Power tailed ruin

Kalashnikov V., Norberg R. Power tailed ruin

work page
[11]

On the ruin probabilities in a

Nyrhinen H. On the ruin probabilities in a

work page
[12]

Finite and infinite time

Nyrhinen H. Finite and infinite time

work page
[13]

Paulsen, J.: Risk theory in stochastic economic environment. Stoch. Proc. Appl. 46 , 327--361 (1993)

work page 1993
[14]

Lecture Notes, Univ

Paulsen, J., Stochastic Calculus with Applications to Risk Theory. Lecture Notes, Univ. of Bergen and Univ. of Copenhagen (1996)

work page 1996
[15]

Paulsen J.: Sharp conditions for certain ruin in a risk process with stochastic return on investments. Stoch. Proc. Appl. 75 , 135--148 (1998)

work page 1998
[16]

Paulsen, J., Gjessing, H.K.: Ruin theory with stochastic return on investments. Adv. Appl. Probab. 29 , 965--985 (1997)

work page 1997
[17]

Ruin probability in the presence of risky investments

Pergamenshchikov S., Zeitouni O. Ruin probability in the presence of risky investments. Stoch. Process. Appl. , 116 (2006), 267--278

work page 2006

[1] [1]

Albrecher, H., Constantinescu, C., Thomann, E.: Asymptotic results for renewal risk models with risky investments. Stoch. Proc. Appl. 122 , 3767--3789 (2012)

work page 2012

[2] [2]

Eberlein, E., Kabanov, Yu., Schmidt, T.: Ruin probabilities for a Sparre Andersen model with investments. Stoch. Proc. Appl. 144 , 72--84 (2022)

work page 2022

[3] [3]

Some mathematical models of risk theory

Frolova A.G. Some mathematical models of risk theory

work page

[4] [4]

Goldie, C.M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 , 126--166 (1991)

work page 1991

[5] [5]

Springer, Berlin (1990)

Grandell, I.: Aspects of Risk Theory. Springer, Berlin (1990)

work page 1990

[6] [6]

Finance Stoch

Kabanov, Yu., Pergamenshchikov, S.: In the insurance business risky investments are dangerous: the case of negative risk sums. Finance Stoch. 20 , 355--379 (2016)

work page 2016

[7] [7]

Finance Stoch

Kabanov, Yu., Pergamenshchikov, S.: Ruin probabilities for a L\'evy-driven generalised Ornstein--Uhlenbeck process. Finance Stoch. 24 , 39--69 (2020)

work page 2020

[8] [8]

Kabanov, Yu., Pukhlyakov, N.: Ruin probabilities with investments: smoothness, IDE and ODE, asymptotic behavior. J. Appl. Probab. 59 , 556--570 (2020)

work page 2020

[9] [9]

Ruin probability under random

red Kalashnikov V., Norberg R. Ruin probability under random

work page

[10] [10]

Power tailed ruin

Kalashnikov V., Norberg R. Power tailed ruin

work page

[11] [11]

On the ruin probabilities in a

Nyrhinen H. On the ruin probabilities in a

work page

[12] [12]

Finite and infinite time

Nyrhinen H. Finite and infinite time

work page

[13] [13]

Paulsen, J.: Risk theory in stochastic economic environment. Stoch. Proc. Appl. 46 , 327--361 (1993)

work page 1993

[14] [14]

Lecture Notes, Univ

Paulsen, J., Stochastic Calculus with Applications to Risk Theory. Lecture Notes, Univ. of Bergen and Univ. of Copenhagen (1996)

work page 1996

[15] [15]

Paulsen J.: Sharp conditions for certain ruin in a risk process with stochastic return on investments. Stoch. Proc. Appl. 75 , 135--148 (1998)

work page 1998

[16] [16]

Paulsen, J., Gjessing, H.K.: Ruin theory with stochastic return on investments. Adv. Appl. Probab. 29 , 965--985 (1997)

work page 1997

[17] [17]

Ruin probability in the presence of risky investments

Pergamenshchikov S., Zeitouni O. Ruin probability in the presence of risky investments. Stoch. Process. Appl. , 116 (2006), 267--278

work page 2006