Asymptotics of survival probabilities and lower tail probability problem

Sergei Levendorski\u{i}; Svetlana Boyarchenko

REVIEW 2 major objections 3 minor 45 references

Asymptotics of survival probabilities and lower tail probability problem

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proves that for Stieltjes-Lévy processes, as $t\to\infty$ the survival probability $P[\bar X_t<x]$ is $1-p_\infty(x)+O(t^{-1})$ when $\mu_1<0$, $p_{\infty,0}(x)t^{-1/2}+O(t^{-1})$ when $\mu_1=0$, and $O(t^{-1})$ when $\mu_1>0$…

desk verdict Explicit SL-process asymptotics are a real contribution, but the proof of the O(t^{-1}) rates has a genuine contour-deformation gap near q=0 that needs fixing. read the letter →

arxiv 2501.04218 v1 pith:2P3DFU4L submitted 2025-01-08 math.PR

Svetlana Boyarchenko , Sergei Levendorskiĭ This is my paper · ORCID

classification math.PR MSC 60G5160G5260-0865C0591G0591G2097M30

keywords survivalprobabilitylowertailWiener-HopffactorizationStieltjes-LévyprocessesSINH-regularLévyasymptoticexpansionspersistenceprobabilities

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Lévy processes model asset prices, insurance reserves, and other quantities whose jumps arrive randomly in time; a basic question is the survival probability $P[\bar X_t < x]$ that the running maximum stays below a fixed level $x$ for a long time. This paper derives the leading large-time asymptotics of this probability for Stieltjes-Lévy (SL) processes, a broad class that includes most popular Lévy models. The answer is controlled by the sign of the first instantaneous moment $\mu_1$: a downward drift gives $P[\bar X_t < x] = 1-p_\infty(x)+O(t^{-1})$, zero drift gives a $t^{-1/2}$ decay with explicit prefactor $p_{\infty,0}(x)$, and upward drift gives $O(t^{-1})$. The coefficients are expressed through the absolutely continuous components of the Stieltjes-Lévy measures and the zeros of the characteristic exponent. For the companion lower tail problem (fixed $t$, $x\downarrow 0$), the paper gives $P[\bar X_t < x] = \kappa_k(t)x^{\nu_+}+O(x^{\nu_++s})$ for SINH-regular processes, with the exponent $\nu_+$ and prefactor $\kappa_k(t)$ determined by the asymptotic decay of the positive Wiener-Hopf factor.

What carries the argument

The machinery is the contour-integral representation of the survival probability together with the Wiener-Hopf factorization identity. For a Lévy process with characteristic exponent $\psi$, the probability is written as a double contour integral; the paper deforms the $q$-contour to $\operatorname{Re} q=\sigma/t$, uses integration by parts to show the large $|\operatorname{Im} q|$ part is $O(t^{-1})$, and then studies $q$ near $0$. The small-$q$ behavior of the positive Wiener-Hopf factor $\varphi_q^+$ is captured through a modified factor $\varphi_q^{-,0}$ defined by (6.13), for which the paper proves uniformity in $q$ near the origin and an explicit representation (6.15) in terms of zeros of $\psi$ and the Stieltjes-Lévy measure $G_-$. The link between $\psi$ and the Lévy measure is Theorem 3.16, which expresses the jump density as an integral of $\operatorname{Im}\psi$ on the boundary of the strip of analyticity; this is what turns the abstract asymptotic coefficients into explicit SL-measure integrals.

What would settle it

Take a KoBoL process with $\mu_1<0$ and numerically evaluate $P[\bar X_t<x]$ for large $t$ by inverting the Fourier-Laplace representation with high precision; if the difference from $1-p_\infty(x)$ is not $O(t^{-1})$ or has a different constant than (6.16), the claim fails. A more direct check is to search the strip $S[\omega'_{-,0}]$ for zeros of $q+\psi(\xi)$ with $\operatorname{Re} q=\sigma/t$ and $|\operatorname{Im} q|\ge \epsilon$: any zero invalidates the integration-by-parts step.

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Extended reading notes

Core claim

The central discovery is that the large-time survival probability for an SL-process is asymptotically determined by local behavior of the characteristic exponent at zero and by the absolutely continuous component of the Stieltjes-Lévy measure. Specifically, if $\mu_1<0$, then $P[\bar X_t<x] = 1-p_\infty(x)+O(t^{-1})$ with $p_\infty(x)$ given by (6.16); if $\mu_1=0$, then $P[\bar X_t<x] = p_{\infty,0}(x)t^{-1/2}+O(t^{-1})$ with $p_{\infty,0}(x)$ given by (6.20); and if $\mu_1>0$, then $P[\bar X_t<x] = O(t^{-1})$. The coefficients $p_\infty(x)$ and $p_{\infty,0}(x)$ are expressed in terms of the density of the absolutely continuous component of the Stieltjes-Lévy measure $G_+$, the modified Wiener-Hopf factor $\varphi_0^{-,0}$, the first two instantaneous moments $\mu_1,\mu_2$, and the zeros of the characteristic exponent on the imaginary axis. The proof works by deforming the $q$-integration contour in the Laplace-Fourier inversion formula, isolating the small-$q$ contribution, and expanding the Wiener-Hopf factors in terms of the roots of $q+\psi(\xi)=0$ and the SL-measures. The lower tail problem is solved at the same level of generality: for SINH-regular processes, $P[\bar X_t<x] = \kappa_k(t)x^{\nu_+}+O(x^{\nu_++s})$ as $x\downarrow 0$, where $\nu_+$ is the decay rate of the positive Wiener-Hopf factor and $\kappa_k(t)$ is a Laplace inversion integral.

Load-bearing premise

The argument assumes that $q+\psi(\xi)$ has no zeros on the entire deformed contour $\operatorname{Re} q=\sigma/t$ with $|\operatorname{Im} q|\ge \epsilon$ inside the strip $S[\omega'_{-,0}]$; the paper verifies this only in a small ball around $q=0$, so an unnoticed zero elsewhere would spoil the error estimate.

Editorial extensions

If this is right

For SL-processes with $\mu_1<0$, survival to a fixed level $x$ converges to a positive limit $1-p_\infty(x)$ at rate $1/t$, giving a first-order correction to ruin probabilities in insurance and barrier option prices.
At zero drift, the survival probability decays as $t^{-1/2}$ with an explicit prefactor, so the persistence exponent is $1/2$ and the constant can be computed from the Lévy measure.
For positive drift, survival up to a fixed level is $O(t^{-1})$, so the probability of never exceeding $x$ is asymptotically small at least at that rate.
The lower tail result gives $P[\bar X_t<x] \approx \kappa(t)x^{\nu_+}$ for small $x$, so the exponent $\nu_+$ and prefactor $\kappa(t)$ determine the near-barrier behavior of no-touch options for any fixed maturity.
The explicit formulas allow direct computation of asymptotics once the Stieltjes-Lévy representation is available, without solving the Wiener-Hopf equation numerically.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The zero-drift $t^{-1/2}$ prefactor depends on $\psi$ only through $\mu_2$ and $\varphi_0^{-,0}$, suggesting a universality class: processes with equal $\mu_2$ and matched small-$x$ structure share the same leading persistence rate, which could be checked against simulations.
Because the paper proves zero-freeness of $q+\psi(\xi)$ only near $q=0$, one could numerically search for stray zeros on the deformed contour; if found for a particular SL-process, the $O(t^{-1})$ error rate would be replaced by a different rate, and the formulas would need modification.
The same small-$q$ Wiener-Hopf expansion may apply to other extremal functionals, such as the joint law of the running maximum and the process value, opening a route to asymptotic joint densities.
For mixtures of stable processes and regular SL-processes, the paper notes the survival problem is open; one could attempt a rescaling argument to reduce it to the lower tail problem, as suggested in Section 6.1.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear [Colleague],

Quick take: this is a serious addendum to the Boyarchenko–Levendorskiĭ SL-process program, and its explicit coefficient formulas are the real contribution. The main soft spot is the proof of the O(t^{-1}) rates in §6.2, where the deformed q-contour leaves the region where zero-free behavior of q+ψ(ξ) is established. That gap is real and needs referee attention, but the core formulas look right and reduce correctly to Brownian motion.

What's new: Proposition 4.2 gives Wiener-Hopf factor representations in terms of SL-measure supports and zeros of q+ψ(ξ). Theorems 6.5 and 6.8 express the survival probability asymptotics through the first instantaneous moments, zeros of ψ, and integrals against the absolutely continuous components of the SL-measures. These are explicit, derived rather than fitted, and they specialize to known Brownian motion results (the t^{-1/2} coefficient with μ2 in (6.20) is exactly right). That is real credit.

The soft spots: the proof that the integrals over |Im q| ≥ ε on the contour Re q = σ/t are O(t^{-1}) is sketched. It says 'integrating by parts and using (6.9)', but (6.9) is only a pointwise bound on derivatives of (q+ψ(ξ))^{-1}; it does not by itself justify uniform bounds on the tails of the contour, and Lemma 6.2 only controls zeros of q+ψ(ξ) for q in B(0,ε). For complex q with large imaginary part, zeros need not stay on iR, and crossing one would add a residue and change the rate. The stress-test note correctly identifies this as the load-bearing gap; it affects Theorems 6.1, 6.6, 6.7 and the coefficient formulas. It's a fixable gap if the authors can establish a zero-free strip for Re q = σ/t with |Im q| ≥ ε, or prove the tail integrals decay by a different argument. I would not call the central claim false — the formulas are too well-behaved and reduce correctly — but the rate proof as written is incomplete.

The positive drift case, μ1 > 0, is stated as P[X̄_t < x] = O(t^{-1}); that's a rate, not a leading-term formula, and calling it 'leading term' in Section 6.1 is an overstatement. Minor.

Who this is for: specialists in Lévy process fluctuation theory and barrier option pricing who use SINH-acceleration or SL-processes. A serious referee should be sent this; the gap is specific and checkable, and the authors have the tools to fix it.

Recommendation: send to peer review, but flag the §6.2 contour-deformation gap as a major comment, not a desk-reject issue.

Referee Report

2 major / 3 minor

Summary. The paper derives asymptotic formulas for the survival probability P[\bar X_t < x] as t→∞ for fixed x>0, for Lévy processes with exponentially decaying tails, and refines the coefficients for Stieltjes-Lévy (SL) processes in terms of SL-measures and zeros of the characteristic exponent. It also reformulates lower tail probability asymptotics as x↓0 for SINH-regular processes. The main results are (6.1)–(6.3), with explicit expressions for p∞(x) and p∞,0(x) in (6.16) and (6.20), obtained by contour deformation and residue calculus.

Significance. If the proofs are completed, the paper gives explicit, parametric formulas for the leading terms of survival probabilities, expressed in terms of objects that are often explicitly computable for popular Lévy models (KoBoL, NIG, NTS, VG). A strong point is that the coefficient formulas (6.16) and (6.20) reproduce known Brownian motion results when specialized, providing a consistency check. The paper is an addendum to the authors' prior work, but the survival-probability asymptotics for SL-processes appear to be new.

major comments (2)

[§6.2, after (6.9)] The O(t^{-1}) estimate for the integrals over {q: Re q = σ/t, |Im q| ≥ ε} uses the bound (6.9) on the entire deformed contour. Lemma 6.2 establishes a lower bound for |q + ψ(ξ)| only for q in a small ball B(0,ε) and ξ in a small strip S(-ω',ω') around 0; it says nothing about the line Re q = σ/t when |Im q| ≥ ε, where q is not small. For an SL-process, the zero-location statement of Proposition 4.2 applies to real q > 0, so for complex q with Re q = σ/t > 0 and |Im q| ≥ ε there may be zeros of q + ψ(ξ) inside the strip S[ω'_-,0]. If such a zero is crossed during the deformation, the residue calculus in (6.10), (6.17) and (6.18) omits a contribution, and the claimed O(t^{-1}) rate could fail or pick up a t^{-1/2} term, which would alter the leading coefficient in (6.1) and (6.3). The authors should prove a global zero-free region for the deformed contour, or otherwise avoid deforming over the full line.
[§6.2, 'Integrating by parts'] The derivation of the O(t^{-1}) rate by integration by parts is only sketched. For each of the integrals in (6.4), (6.6) and (6.8), one needs to identify the boundary terms as |Im q| → ∞ and justify that they vanish or are negligible, and one needs uniform estimates on the q-derivative of the entire integrand, including the inner ξ-integral and the factor φ_q^+(ξ). Without these estimates, the statement 'Integrating by parts and using (6.9), we obtain' is not a complete proof. Since the O(t^{-1}) accuracy is part of the theorems' claims, the details should be supplied or a reference given where these estimates appear.

minor comments (3)

[Theorem 6.8] Theorem 6.8 states that (6.1) holds with p∞(x), but the case μ1 = 0 should correspond to (6.3) with p∞,0(x); the displayed formula is the coefficient of the t^{-1/2} term.
[§6.2, after (6.8)] The notation 'S(ω'_- ,0]' should be 'S[ω'_-,0]' for consistency with the rest of the paper.
[§2.2.4] The abbreviations 'VGP' and 'VG' are used interchangeably; using one consistent label would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the asymptotic coefficients are derived by contour deformation from integral representations, and prior SL-process results are used as theorems, not as fitted inputs.

full rationale

The paper's central claims (6.16) and (6.20) are obtained by substituting analytic representations of the Wiener-Hopf factors into the Laplace-Fourier inversion formulas (6.4), (6.6), (6.8), then deforming contours and applying residue calculus. The coefficients p_infty(x) and p_infty,0(x) are expressed in terms of the characteristic exponent psi, its zeros, and the SL-measures G_±; the latter are themselves defined from psi via Theorem 3.16(b), so the final formulas are explicit rewritings of the same model ingredients, not independently fitted quantities. No parameter is calibrated to the survival probability being predicted, and no 'prediction' is obtained by substituting a quantity that was fit to that same target. The paper relies heavily on the authors' previous papers [11, 12, 13, 18] for definitions of RLPE, SINH-regular and (s)SL processes and for integral representations of Wiener-Hopf factors; this is normal use of prior mathematical work. The load-bearing zero-location statement for SL-processes, 'for q > 0, the equation q + psi(xi) = 0 has no zeros outside the imaginary axis iR' (attributed to [18], Section 1), is a prior theorem with stated assumptions that do not include the target asymptotic result; citing it does not make the present derivation circular. The genuine weakness identified in the proof is the claim in Section 6.2 that the integrals over |Im q| >= epsilon are O(t^{-1}) using (6.9), which requires q + psi(xi) to be zero-free on the whole deformed contour, whereas Lemma 6.2 only establishes the zero structure for q in a small ball around 0. This is a correctness gap in the proof, not a circularity, because even if the gap is filled, the derivation is still from the characteristic exponent and SL-measures rather than from the quantity being predicted. The paper also explicitly notes in Remark 4.3.1 that for sSL-processes zeros may lie outside iR and a proof is not supplied; again, this is a stated limitation, not a circular step. No step in the derivation reduces by construction to its own inputs, and no fitted input is renamed as a prediction. Score 0 is therefore appropriate.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new free parameters or entities. It relies on the standard Stieltjes-Lévy framework from the authors' prior work, plus several technical assumptions that are either part of the model class or needed for the contour deformation arguments.

assumptions (6)

standard math Standard complex analysis: Cauchy integral formula, analytic continuation, residue theorem.
Used throughout for contour deformations and Laplace-Fourier inversion in Sections 4-6.
domain assumption The process X is an SL-process (or sSL/SINH-regular), so ψ is analytic in a strip and satisfies the asymptotic conditions of Definition 2.1.
This is the class of processes for which the results are stated; see Definitions 2.1 and 3.14.
domain assumption The integral representation (6.4)-(6.6) holds for Lévy processes with ψ analytic in a strip (cited from [7]).
The starting point for all asymptotic derivations in Section 6.
ad hoc to paper There exists ω'_- < 0 such that q + ψ(ξ) ≠ 0 for all q on the deformed contour Re q = σ/t and all ξ in S[ω'_-,0].
Needed for contour deformation without crossing zeros; not proven for |Im q| ≥ ε. This is the weakest point of the proof.
ad hoc to paper Technical integrability condition (b) in Theorems 6.5 and 6.8: ∫_{U+} |ψ(-iw-0)|^{-2} G+(dw) < ∞.
Imposed to ensure convergence of the contour integrals after deformation to the cut.
domain assumption Zeros of ψ on i(-∞,0) are simple.
Stated in Theorems 6.5 and 6.8 to simplify the residue sums.

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Pith. "Pith review of Asymptotics of survival probabilities and lower tail probability problem." pith.science (2026). https://pith.science/paper/2P3DFU4L

@misc{pith2026250104218,
  author       = {Pith},
  title        = {Pith review of: Asymptotics of survival probabilities and lower tail probability problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2P3DFU4L}},
  note         = {Machine review of arXiv:2501.04218}
}

abstract

The present paper is an addendum to the paper ``L\'evy models amenable to efficient calculations", where we introduced a general class of Stieltjes-L\'evy processes (SL-processes) and signed SL processes defined in terms of certain Stieltjes-L\'evy measures. We demonstrated that SL-processes enjoyed all properties that we used earlier to develop efficient methods for evaluation of expectations of functions of a L\'evy process and its extremum processes, and proved that essentially all popular classes of L\'evy processes are SL-processes; sSL-processes fail to possess one important property. In the present paper, we use the properties of (s)SL-processes to derive new formulas for the Wiener-Hopf factors $\phi^\pm_q$ for small $q$ in terms of the absolute continuous components of SL-measures and their densities, and calculate the leading terms of the survival probability also in terms of the absolute continuous components of SL-measures and their densities. The lower tail probability is calculated for more general classes of SINH-regular processes constructed earlier.

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