{"id":"2570c380-3cf1-498a-8f30-2f6f2a84f877","arxiv_id":"2501.04218","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors compute leading-order asymptotic coefficients for survival and lower-tail probabilities of Stieltjes-Lévy processes.","lead":"This paper derives asymptotic formulas for the probability that a Lévy process with jumps stays below a fixed level over time. It expresses the leading constants in terms of the process's characteristic exponent and a Stieltjes-Lévy measure, covering many finance models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Contour deformation in §6.2 needs q+ψ zero-free on the whole line Re q=σ/t, but Lemma 6.2 only proves this for q in a small ball around 0; uncontrolled zeros outside that ball could break the O(t^{-1}) tail bound.","rationale":"The paper's central contribution is the asymptotics (6.1)-(6.3) with the coefficient formulas (6.16) and (6.20). The proof strategy is uniform: deform the q-contour from Re q = σ0 to Re q = σ/t, estimate the far tails by integration by parts, and then analyze the local piece using Lemma 6.2. The far-tail estimate is the linchpin: without a zero-free region for q+ψ on the whole deformed contour, the integration-by-parts bound (6.9) has no uniform justification, and any missed pole can contribute a term at least as large as the claimed error. Lemma 6.2 only supplies local information near q=0, and the earlier existence statement of a strip free of q+ψ ∈ (-∞,0] for fixed σ0 > 0 does not automatically extend as σ0 → 0 and |Im q| → ∞. The reader's verdict of CONDITIONAL is therefore appropriate: the asymptotic formulas are plausible and are corroborated by the Brownian specialization and by the internal structure of the derivation, but the proof as written has a real gap at a load-bearing step. The proposed numerical/analytic check on an explicit SL model would settle whether the gap is merely a missing lemma or reflects a genuine failure of the O(t^{-1}) rates. I agree with the reader's weakest_assumption and see no reason to change the verdict.","tokens_in":25621,"tokens_out":11103,"duration_ms":113847,"concrete_test":"For a concrete SL-process with explicit ψ (e.g., CGMY with parameters c=1, ν=1.5, λ_-=-2, λ_+=3, chosen so that μ1<0), fix σ=1 and t=10^3, and use the argument principle on a fine grid in S[ω'_-,0] to count zeros of q+ψ(ξ) for q=σ/t+iy with |y|∈[ε,10^3]. If no zero appears, repeat for several parameter sets and also for ν∈(0,1) and ν=2; absence of zeros is consistent with the proof gap being cosmetic. If a zero is found, evaluate the omitted residue contribution to (6.4) numerically and compare P[\\bar X_t<x] with the RHS of (6.1); a nonzero residue contribution disproves the O(t^{-1}) claim, whereas a zero or exponentially small contribution supports the theorem as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The O(t^{-1}) estimates for the integrals over |Im q| ≥ ε in §6.2 are obtained by integrating by parts in q after invoking the bound (6.9). That bound presupposes that q+ψ(ξ) is uniformly bounded away from zero on the entire deformed contour {Re q = σ/t} and on the strip S[ω'_-,0]. Lemma 6.2, however, only establishes the local structure of zeros for q in B(0,ε) and a lower bound for q in that ball; for |Im q| ≥ ε, with q = σ/t + iy and y large, the paper gives no argument excluding zeros of q+ψ(ξ) off the imaginary axis. For complex q with Re q > 0, zeros need not remain on iR even for SL-processes, since the statements in Proposition 4.2 are for real q > 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. This gap directly affects Theorems 6.1, 6.6, 6.7 and the SL refinements (6.16) and (6.20). The reader's weakest assumption identifies exactly this point; I found no separate flaw that is more load-bearing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":25992,"tokens_out":9986,"duration_ms":86413,"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":[{"comment":"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.","section":"§6.2, after (6.9)"},{"comment":"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.","section":"§6.2, 'Integrating by parts'"}],"minor_comments":[{"comment":"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.","section":"Theorem 6.8"},{"comment":"The notation 'S(ω'_- ,0]' should be 'S[ω'_-,0]' for consistency with the rest of the paper.","section":"§6.2, after (6.8)"},{"comment":"The abbreviations 'VGP' and 'VG' are used interchangeably; using one consistent label would improve readability.","section":"§2.2.4"}],"recommendation":"major_revision","confidential_remarks":"The main formulas are likely correct, and the paper is an incremental but useful addendum to the authors' previous work. The proof gap in Section 6.2 is significant because it affects the error terms of all three theorems. If the authors can close the gap with a lemma on the zero-free region and supply the omitted estimates, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nQuick 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.\n\nWhat'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.\n\nThe 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.\n\nThe 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.\n\nWho 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.\n\nRecommendation: send to peer review, but flag the §6.2 contour-deformation gap as a major comment, not a desk-reject issue.","headline":"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.","tokens_in":26412,"tokens_out":5579,"would_cite":true,"duration_ms":54516,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G51","60G52","60-08","65C05","91G05","91G20","97M30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $\\mu_1>0$…","keywords":["survival probability","lower tail probability","Wiener-Hopf factorization","Stieltjes-Lévy processes","SINH-regular Lévy processes","Lévy processes","asymptotic expansions","persistence probabilities"],"falsifier":"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.","tokens_in":25420,"feed_emoji":"📉","tokens_out":9209,"duration_ms":76479,"temperature":0.7,"pith_summary":"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.","feed_headline":"Lévy survival asymptotics: 1/t decay, 1/√t at zero drift","feed_subtitle":"New explicit formulas give the leading large-time survival probability for Stieltjes-Lévy processes in terms of their jump measures.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"defines Stieltjes-Lévy processes and proves the zero-free property of $q+\\psi(\\xi)$ outside the imaginary axis that underpins the small-$q$ expansions.","marker":"[18]"},{"why":"supplies the integral representations (6.4)-(6.5) for survival probabilities and the analytic continuation formulas for Wiener-Hopf factors.","marker":"[12]"},{"why":"derives the contour integral formulas (4.2)-(4.5) for the Wiener-Hopf factors used throughout.","marker":"[13]"},{"why":"extends the Laplace-Fourier inversion formulas to all Lévy processes with characteristic exponent analytic in a strip, justifying the general asymptotic framework.","marker":"[7]"},{"why":"introduces SINH-regular processes and their asymptotic cone conditions used in the lower tail probability problem.","marker":"[14]"},{"why":"provides the barrier-option near-barrier asymptotics reformulated here as the lower tail probability theorem.","marker":"[6]"},{"why":"gives the Carr-randomization convergence and lower tail asymptotic formulas used in Theorem 5.1.","marker":"[34]"}],"fun_headline_variants":["SL-process survival: zero-drift tail ∝ t^-1/2, else O(t^-1)","Lévy survival: leading terms from SL-measure densities","New tails for Lévy: survival and lower-tail leading order","Survival asymptotics for SL-processes: explicit leading coefficients","SINH-regular lower tails: explicit x^ν_+ leading term"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["SL-process survival: zero-drift tail ∝ t^-1/2, else O(t^-1)","Lévy survival: leading terms from SL-measure densities","New tails for Lévy: survival and lower-tail leading order","Survival asymptotics for SL-processes: explicit leading coefficients","SINH-regular lower tails: explicit x^ν_+ leading term"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001446,"raw_usage":{"total_tokens":5918,"prompt_tokens":1132,"completion_tokens":4786,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":748,"completion_tokens_details":{"reasoning_tokens":4687}},"tokens_in":748,"tokens_out":4786,"duration_ms":32156,"temperature":1.0,"reasoning_tokens":4687,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:40:57.787091+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Boyarchenko and S","cited_arxiv_id":null,"evidence_quote":"defines Stieltjes-Lévy processes and proves the zero-free property of $q+\\psi(\\xi)$ outside the imaginary axis that underpins the small-$q$ expansions."},{"cited_title":"Boyarchenko and S","cited_arxiv_id":null,"evidence_quote":"supplies the integral representations (6.4)-(6.5) for survival probabilities and the analytic continuation formulas for Wiener-Hopf factors."},{"cited_title":"Boyarchenko and S","cited_arxiv_id":null,"evidence_quote":"derives the contour integral formulas (4.2)-(4.5) for the Wiener-Hopf factors used throughout."},{"cited_title":"Boyarchenko and S","cited_arxiv_id":null,"evidence_quote":"extends the Laplace-Fourier inversion formulas to all Lévy processes with characteristic exponent analytic in a strip, justifying the general asymptotic framework."},{"cited_title":"Boyarchenko and S","cited_arxiv_id":null,"evidence_quote":"introduces SINH-regular processes and their asymptotic cone conditions used in the lower tail probability problem."},{"cited_title":"Boyarchenko, M","cited_arxiv_id":null,"evidence_quote":"provides the barrier-option near-barrier asymptotics reformulated here as the lower tail probability theorem."},{"cited_title":"Levendorski ˘i","cited_arxiv_id":null,"evidence_quote":"gives the Carr-randomization convergence and lower tail asymptotic formulas used in Theorem 5.1."}],"review_version":1}