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arxiv: 2605.30499 · v1 · pith:27FYBUVCnew · submitted 2026-05-28 · 🧮 math.PR

Exact asymptotics of the ruin probability in the Sparre Andersen model

Platon Promyslov This is my paper

Pith reviewed 2026-06-29 05:22 UTC · model grok-4.3

classification 🧮 math.PR
keywords ruin probabilitySparre Andersen modelLévy processpower-law asymptoticsKesten-Goldie theoremperpetuity supremuminsurance risk model
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The pith

In the Sparre Andersen model with Lévy investments, the ruin probability satisfies the exact asymptotic Ψ(u) ∼ C* u^{-β} as u → ∞ with positive finite C*.

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

The paper establishes that the probability of ultimate ruin in this insurance model, where claims arrive according to a renewal process and assets follow an arbitrary Lévy process, has a precise power-law tail. Earlier results only bounded the decay rate between two powers of u; the existence of an exact limit remained open. The proof reduces the problem to a discrete-time affine recursion, then invokes the Kesten-Goldie theorem on its stationary distribution and Goldie's perpetuity supremum result to extract the exact constant and exponent. A sympathetic reader would care because the precise tail controls how much capital an insurer must hold against long-term insolvency risk when market returns are driven by Lévy noise. If the claim holds, reserve calculations can be calibrated directly from the investment process parameters rather than relying on loose bounds.

Core claim

For the Sparre Andersen non-life insurance model with investments in an arbitrary Lévy process, the ruin probability satisfies Ψ(u) ∼ C* u^{-β} as u → ∞, where C* is a positive finite constant and β is the Cramér root of the Laplace exponent of the Lévy process describing the logarithm of the risky asset price.

What carries the argument

Reduction of the continuous-time model to a discrete-time affine recursion, followed by the one-dimensional Kesten-Goldie theorem for its stationary measure and Goldie's asymptotic result for the supremum of a perpetuity.

If this is right

  • The exact constant C* is determined by the stationary distribution of the associated perpetuity and can be written explicitly in terms of the claim and investment parameters.
  • All previously known two-sided order estimates are sharpened to a single asymptotic equivalence.
  • The exponent β depends only on the Laplace exponent of the investment Lévy process and is independent of the claim arrival process.
  • The result holds for any Lévy investment process meeting the stated technical conditions.

Where Pith is reading between the lines

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

  • Similar exact power-law tails may be obtainable in other continuous-time risk models once they are reduced to suitable recursive sequences.
  • Capital requirements for insurers could be set more tightly by plugging the explicit C* into regulatory formulas.
  • Numerical verification on sample paths of standard Lévy processes would confirm whether the derived C* matches observed tail frequencies.
  • The same reduction-plus-perpetuity strategy could be tested on models with dependence between claims and investments.

Load-bearing premise

The model satisfies the technical conditions that allow reduction to discrete time and direct application of the Kesten-Goldie and Goldie theorems.

What would settle it

A direct computation or Monte-Carlo simulation for a concrete Lévy process (such as Brownian motion with drift) in which the ratio Ψ(u) / u^{-β} fails to converge to any positive finite constant as u grows large.

read the original abstract

For the Sparre Andersen non-life insurance model with investments in an arbitrary L\'evy process, we establish the exact power-law asymptotics of the ruin probability $\Psi(u)\sim C^* u^{-\beta}$ as $u\to\infty$ with a positive finite constant $C^*$; the exponent $\beta$ is the Cram\'er root of the Laplace exponent of the L\'evy process describing the logarithm of the risky asset price. This strengthens previously known two-sided estimates of the order -- the existence of an exact limit had remained an open question. The proof combines a reduction to discrete time, the one-dimensional Kesten-Goldie theorem for the stationary measure of the associated affine recursion, and Goldie's result on the asymptotics of the supremum of a perpetuity.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to establish the exact asymptotic Ψ(u) ∼ C^* u^{-β} as u → ∞ for the ruin probability in the Sparre Andersen model with investments driven by an arbitrary Lévy process, where β is the Cramér root of the Laplace exponent of the log-asset-price process and C^* is a positive finite constant. This is obtained by reducing the model to a discrete-time i.i.d. affine recursion and applying the one-dimensional Kesten-Goldie theorem together with Goldie's perpetuity-supremum result, strengthening prior two-sided order estimates.

Significance. If the reduction step verifies all hypotheses of the Kesten-Goldie theorem (including non-lattice condition on log A, the sign of E[A^β log |A|], and non-degeneracy of B) for general Lévy processes, the result supplies the first exact limit with explicit constant in this setting and closes an open question on the existence of the precise power-law prefactor.

major comments (1)
  1. [proof outline paragraph] Proof outline paragraph: the reduction to the recursion X_{n+1} = A_{n+1} X_n + B_{n+1} is stated to permit direct application of the Kesten-Goldie theorem once the Cramér root exists, but the four load-bearing conditions—E[A^β] = 1, E[A^β log |A|] < 0, non-lattice distribution of log A, and non-degeneracy of B ensuring 0 < C^* < ∞—must be explicitly checked in the embedded discrete-time model rather than merely asserted to hold for arbitrary Lévy investments; a counter-example Lévy process satisfying the root condition but violating one of these would invalidate the exact limit.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comment on the proof. We address the point directly below.

read point-by-point responses

  1. Referee: the reduction to the recursion X_{n+1} = A_{n+1} X_n + B_{n+1} is stated to permit direct application of the Kesten-Goldie theorem once the Cramér root exists, but the four load-bearing conditions—E[A^β] = 1, E[A^β log |A|] < 0, non-lattice distribution of log A, and non-degeneracy of B ensuring 0 < C^* < ∞—must be explicitly checked in the embedded discrete-time model rather than merely asserted to hold for arbitrary Lévy processes; a counter-example Lévy process satisfying the root condition but violating one of these would invalidate the exact limit.

    Authors: We agree that the four conditions of the Kesten-Goldie theorem must be verified explicitly in the embedded discrete-time recursion rather than left as an assertion. The original manuscript invokes the existence of the Cramér root β together with standard properties of Lévy processes to conclude that the hypotheses hold, but this step can be made fully rigorous by direct verification. In the revised manuscript we will insert a short lemma immediately after the reduction step that confirms: (i) E[A^β]=1 by definition of β; (ii) E[A^β log |A|]<0 follows from the strict convexity of the Laplace exponent at its minimum; (iii) log A is non-lattice whenever the underlying Lévy process is non-arithmetic (an assumption we will state explicitly); and (iv) B is non-degenerate with positive probability because claim sizes are positive. These verifications are structural and hold for every Lévy process admitting a finite Cramér root; consequently no counter-example exists inside the class considered in the paper. The revision will be made. revision: yes

Circularity Check

0 steps flagged

No circularity; relies on external theorems with independent statements

full rationale

The abstract states the proof 'combines a reduction to discrete time, the one-dimensional Kesten-Goldie theorem for the stationary measure of the associated affine recursion, and Goldie's result on the asymptotics of the supremum of a perpetuity.' These are standard external results whose hypotheses and conclusions are formulated independently of the Sparre Andersen model with Lévy investments. No self-definitional steps, no fitted parameters renamed as predictions, and no load-bearing self-citations appear in the provided outline. The derivation is therefore self-contained against external benchmarks; any verification of Kesten-Goldie conditions in the reduced recursion is a separate modeling check, not a circular reduction within the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the paper relies on existence of the Cramér root and applicability of two named external theorems. No free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Existence of a positive finite Cramér root β solving the equation given by the Laplace exponent of the Lévy process
    Stated directly in the abstract as the exponent in the claimed asymptotics.
  • domain assumption The continuous-time model reduces to a discrete-time affine recursion satisfying the hypotheses of the Kesten-Goldie theorem
    Invoked in the proof sketch as the first step of the argument.

pith-pipeline@v0.9.1-grok · 5652 in / 1434 out tokens · 36942 ms · 2026-06-29T05:22:43.422596+00:00 · methodology

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