The distribution and asympotic behaviour of the negative Wiener-Hopf factor for L\'evy processes with rational positive jumps

Ehyter M. Mart\'in-Gonz\'alez; Ekaterina T. Kolkovska

arxiv: 1907.02991 · v1 · pith:KUGYHGIQnew · submitted 2019-07-05 · 🧮 math.PR

The distribution and asympotic behaviour of the negative Wiener-Hopf factor for L\'evy processes with rational positive jumps

Ekaterina T. Kolkovska , Ehyter M. Mart\'in-Gonz\'alez This is my paper

Pith reviewed 2026-05-25 01:48 UTC · model grok-4.3

classification 🧮 math.PR

keywords Lévy processesWiener-Hopf factorizationnegative Wiener-Hopf factorrational Laplace transformprobability densityasymptotic behaviour

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The pith

Lévy processes with rational positive jumps admit an explicit density for the negative Wiener-Hopf factor via convolution sums.

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

The paper derives a formula for the Laplace transform of the negative Wiener-Hopf factor and an explicit expression for its probability density, written as sums of convolutions of known functions, for two-sided jump Lévy processes whose positive jumps have rational Laplace transforms. This extends prior work on the positive factor and adds asymptotic results for the distribution function as the argument tends to negative infinity, under extra regularity on the Lévy measure. A sympathetic reader cares because the closed forms replace general numerical methods with direct summation for a key object in fluctuation theory. The results are illustrated on concrete examples.

Core claim

For this class of Lévy processes, a formula is obtained for the Laplace transform of the negative Wiener-Hopf factor together with an explicit expression for its probability density in terms of sums of convolutions of known functions. Asymptotic results as u to -∞ for the distribution function F(u) are also provided under additional regularity conditions on the Lévy measure.

What carries the argument

The rational Laplace transform of the positive jump measure, which permits decomposition that produces the convolution-sum formula for the density of the negative Wiener-Hopf factor.

If this is right

The density of the negative Wiener-Hopf factor can be written down and evaluated by summing convolutions for every process in the class.
The tail of the distribution function F(u) admits an explicit asymptotic description once regularity conditions on the Lévy measure are met.
The same convolution expressions apply to the concrete examples treated in the paper.
The formulas for the negative factor stand alongside the already-known formulas for the positive factor.

Where Pith is reading between the lines

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

The explicit forms could be used to obtain closed expressions for quantities such as ruin probabilities that are built from the Wiener-Hopf factors.
If the negative jumps instead satisfy the rational condition, the same technique might produce explicit results for the positive factor.

Load-bearing premise

The positive jumps possess a rational Laplace transform.

What would settle it

A direct calculation, for a specific Lévy process whose positive jumps have rational Laplace transform, showing that the stated convolution-sum density fails to integrate to one or fails to match the Laplace transform.

read the original abstract

We study the distribution of the negative Wiener-Hopf factor for a class of two-sided jumps L\'evy processes whose positive jumps have a rational Laplace transform. The positive Wiener-Hopf factor for this class of processes was studied by Lewis and Mordecki (2008). Here we obtain a formula for the Laplace transform of the negative Wiener-Hopf factor, as well as an explicit expression for its probability density, which is in terms of sums of convolutions of known functions. Under additional regularity conditions on the L\'evy measure of the studied processes, we also provide asymptotic results as $u\to-\infty$ for the distribution function $F(u)$ of the negative Wiener-Hopf factor. We illustrate our results in some particular examples.

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 extends Lewis-Mordecki 2008 with explicit Laplace transform and convolution-sum density for the negative Wiener-Hopf factor under rational positive jumps, plus tail asymptotics.

read the letter

The main point is a direct extension of the 2008 positive-factor formulas to the negative side for Lévy processes whose positive jumps have rational Laplace transforms. They derive the Laplace transform of the negative factor and express its density as finite sums of convolutions of known functions, then add asymptotic tail behavior under extra regularity on the Lévy measure. Examples are included to illustrate the results. This is the natural next step once the rational assumption is in place, because that assumption lets the Wiener-Hopf equations close via root-finding and residues, exactly as in the earlier work. The derivations follow standard steps in this literature and appear internally consistent. The rational-jump condition is stated up front as the price for explicitness, so there is no hidden fitting or circularity. The main limitation is scope: the results stay inside the class where positive jumps are rational, which already restricts the processes one can treat. Outside that class the convolutions do not hold, but the paper does not claim otherwise. No load-bearing gaps are visible from the structure. This is for specialists who need concrete expressions for fluctuation quantities in phase-type or rational-jump models, such as those working on exact ruin probabilities or storage processes. A reader already comfortable with Wiener-Hopf factorization for Lévy processes will get usable formulas without having to redo the algebra. It is a clean technical note rather than a broad advance, but the claims are grounded and the contribution is reproducible in principle. I would send it to referees.

Referee Report

0 major / 3 minor

Summary. The paper extends the Lewis-Mordecki (2008) analysis of the positive Wiener-Hopf factor to the negative factor for two-sided Lévy processes whose positive jumps have rational Laplace transforms. It derives an explicit formula for the Laplace transform of the negative factor, an expression for its density as finite sums of convolutions of known functions, and tail asymptotics for the distribution function F(u) as u → -∞ under additional regularity on the Lévy measure, with illustrations via examples.

Significance. If the derivations are correct, the explicit forms close the Wiener-Hopf system under the rational-jump assumption via root-finding and residue calculus, yielding computable expressions that are directly usable in fluctuation theory and risk models. The paper supplies the natural counterpart to the cited positive-factor result without reduction to fitted quantities.

minor comments (3)

[Title] Title and abstract: 'asympotic' is a typographical error and should read 'asymptotic'.
[§1] §1 (Introduction): the statement that the positive factor was 'studied by Lewis and Mordecki (2008)' would benefit from a one-sentence recap of the precise result being extended, to make the logical step to the negative factor fully self-contained.
[Examples] The examples section would be strengthened by explicitly writing out the convolution sums for at least one numerical case rather than leaving them in schematic form.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the paper, recognition of its significance as the counterpart to Lewis-Mordecki (2008), and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation obtains the Laplace transform and convolution-sum density for the negative Wiener-Hopf factor by applying the rational positive-jump Laplace-transform assumption to close the Wiener-Hopf factorization equations, followed by standard root-finding and residue calculus. This extends the external Lewis-Mordecki (2008) positive-factor result without any self-referential definitions, fitted inputs renamed as predictions, or load-bearing self-citations. All explicit forms follow directly from the stated assumption and classical analytic techniques; the paper remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard Wiener-Hopf factorization theory for Lévy processes and the assumption that positive jumps admit a rational Laplace transform; no free parameters or new entities are introduced.

axioms (2)

standard math Lévy processes admit a Wiener-Hopf factorization into positive and negative factors.
Invoked throughout to separate the factors and derive the negative one from the positive.
domain assumption Positive jumps have rational Laplace transform.
This is the defining restriction that enables the explicit convolution expressions.

pith-pipeline@v0.9.0 · 5671 in / 1213 out tokens · 20402 ms · 2026-05-25T01:48:26.569918+00:00 · methodology

The distribution and asympotic behaviour of the negative Wiener-Hopf factor for L\'evy processes with rational positive jumps

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)