{"id":"d10086ce-d87f-4e6c-88ca-9c4117038353","arxiv_id":"1908.02071","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper presents a Volterra integral equation for OU first passage times that is meant to generalize the Fortet renewal equation, but the derivation step is algebraically incorrect.","lead":"This paper claims to derive a new integral equation for the first passage time density of an Ornstein-Uhlenbeck process, with a kernel that contains a parabolic cylinder function. The central derivation contains a sign error, so the equation does not follow from the stated identities.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (14) does not follow from Eq. (13): the simplification drops the nonzero exponential factor exp(((x-μθ)^2-(S-μθ)^2)/(2σ²θ)), so the central integral equation is false for x>S.","rationale":"The reader's verdict of REJECT is justified. The central equation (14) is not derived correctly: the step from (13) to (14) conflates A and B and drops a constant exponential factor. This is not a matter of convention or a missing factor that can be absorbed by redefining the kernel; it is a concrete algebraic error. The missing factor exp(((x-μθ)^2-(S-μθ)^2)/(2σ²θ)) depends on the threshold S and the observation point x, and it is time-independent. For x>S it is not 1, so Eq. (14) cannot be correct as stated. The q=0 reduction is illuminating: for q=0 the correct convolution theorem produces the Fortet equation multiplied by this same factor, so Eq. (14) is compatible with Fortet only when (x-μθ)^2=(S-μθ)^2. The paper's Section 4 retrieval of the closed form for S=μθ works because this condition holds identically (x=S=μθ), but that is a degenerate case and provides no support for the general equation. The time-dependent equation (17) is asserted by analogy and checked only by an unreported numerical evaluation; given the failure of (14), (17) is unsupported. No machine-checked proof or reproducible code is provided. Therefore the paper's central claim is not established, and the appropriate verdict is rejection. We agree with the reader's weakest-assumption analysis.","tokens_in":9386,"tokens_out":18362,"duration_ms":145435,"concrete_test":"Evaluate the algebraic identity underlying the step from (13) to (14) with σ=θ=1, μ=0, x0=0, x=1, S=2, t=1. Compute R = exp(((x0-μθ)^2-(S-μθ)^2)/(2σ²θ)) * exp(-A²/(2σ²θ(1-e^{-2t/θ}))) / exp(-B²/(2σ²θ(1-e^{-2t/θ}))), where A=(μθ-x0)+(x-μθ)e^{-t/θ} and B=(x-μθ)-(x0-μθ)e^{-t/θ}. The predicted ratio is exp(((x-μθ)^2-(S-μθ)^2)/(2σ²θ)) = e^{-1.5} ≈ 0.223, not 1. Equivalently, solve the Fortet equation (5) numerically for g(S,t|x0), substitute into Eq. (14) with q=0, and observe that the left side is e^{-1.5} times the right side. This would directly falsify the central equation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing claim is that Eq. (14) is a new Volterra integral equation for the first-passage-time density of the Ornstein–Uhlenbeck process. The derivation from (13) to (14) is algebraically wrong. In (13) the inverse-Laplace expression contains exp(((x0-μθ)^2-(S-μθ)^2)/(2σ²θ)) times exp(-A²/(2σ²θ(1-e^{-2t/θ}))) with A=(μθ-x0)+(x-μθ)e^{-t/θ}. Equation (14) replaces this by exp(-B²/(2σ²θ(1-e^{-2t/θ}))) with B=(x-μθ)-(x0-μθ)e^{-t/θ}, dropping the constant prefactor. Since A²-B²=((μθ-x0)²-(x-μθ)²)(1-e^{-2t/θ}), the two exponentials differ by the factor exp(((x-μθ)²-(S-μθ)²)/(2σ²θ)), which is not 1 when x>S. Hence Eq. (14) is generally false, and the time-dependent version (17) inherits the defect. The special case S=μθ used in Section 4 has x=S=μθ, where the missing factor equals 1, so it does not test the general claim. The reported numerical verification of (17) is not reproducible (no code or parameters) and cannot outweigh an explicit algebraic contradiction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a new Volterra integral equation for the first-passage-time density of the Ornstein--Uhlenbeck process. The starting point is the Laplace transform of the first-passage-time density for a constant threshold, which contains a ratio of parabolic cylinder functions. Using inverse Laplace transforms for products of parabolic cylinder functions previously derived by the author, the paper claims to invert the product of this Laplace transform with a suitably chosen factor and thereby to obtain a new integral equation whose kernel contains a parabolic cylinder function. The Fortet renewal equation is recovered when the parabolic cylinder order q is zero, and a Volterra equation of the second kind is obtained for x=S and q=1. The paper further claims that the equation extends to time-dependent thresholds, supported by a numerical check for one exponential threshold, and that the kernel is regular for q≤−1. Section 4 uses the q=1 case to retrieve a known closed-form density for the threshold S=μθ.","tokens_in":9732,"tokens_out":35365,"duration_ms":305968,"significance":"If the proposed integral equation were correct, it would give a new representation of first-passage-time densities for the Ornstein--Uhlenbeck process and could be of numerical interest, especially because the Fortet equation appears as a special case. The paper also aims to unify several known results through a single kernel containing parabolic cylinder functions. However, the central equation is not consistent with the paper's own known closed-form density: at q=1 and x=S=μθ, Eq. (14) yields a result larger by a factor of √2 than the standard formula (19). Since this is a numerical-algebraic contradiction at a parameter point explicitly treated in the paper, the main claim cannot be accepted. Credit is due for the q=0 reduction, which is internally consistent, and for the attempt to connect the inverse Laplace transform literature with first-passage-time equations, but these strengths do not compensate for the incorrect central equation.","major_comments":[{"comment":"The reader's stated objection to the step from (13) to (14) -- that the exponential changes from exp(-((μθ−x0)+(x−μθ)e^{-t/θ})^2/(2σ²θ(...))) to exp(-((x−μθ)−(x0−μθ)e^{-t/θ})^2/(2σ²θ(...))) and that a factor is therefore dropped -- does not by itself invalidate the derivation. When the kernel is rewritten with the threshold appearing in both the exponential and the D_q argument, the relation F_u(t)=(1/θ)exp(((x−μθ)^2−(u−μθ)^2)/(2σ²θ))K_u(t) holds, so the prefactor in (13) cancels against the same prefactor generated on the left-hand side. The equation (14) is nevertheless wrong. Consider q=1 and x=S=μθ. Then the left-hand kernel of (14) vanishes because D_1(0)=0, and (14) reduces to g(μθ,t|x0)=(2/(θ√π))K_{x0}(t), where K_{x0}(t) is the right-hand kernel. Using D_1(z)=z exp(−z²/4), this gives g(μθ,t|x0)=2√2(μθ−x0)/(σ θ^{3/2}√π) e^{-t/θ}(1−e^{-2t/θ})^{-3/2} exp(−(x0−μθ)^2 e^{-2t/θ}/(σ²θ(1−e^{-2t/θ}))). This is √2 times the known formula (19) quoted in the paper. Since (19) is standard and is also the paper's own target, Eq. (14) cannot be correct.","section":"Section 3, Eq. (14)"},{"comment":"The extension from constant to time-dependent thresholds is not derived. After specializing to q=0, the paper notes that the Fortet equation holds for S(t) and then asserts that the same extension holds for q≠0, supported only by the statement that 'numerical evaluation reveals' equality for the exponential threshold (15). No parameter values, quadrature method, error tolerance, or code are provided, so the verification is not reproducible; moreover, because Eq. (14) is false, this numerical check cannot provide evidence for Eq. (17). A proof, or at minimum a fully specified and reproducible numerical experiment, is required before Eq. (17) can be accepted.","section":"Section 3, Eqs. (15)-(17)"},{"comment":"The printed Eq. (20) does not follow from Eq. (14). Carrying out the q=1, x=S inversion in (14) gives a factor √2 in both the inhomogeneous term and the integral kernel, whereas Eq. (20) as printed has coefficient 2 and agrees with the known result (19). Thus Eqs. (14) and (20) are mutually inconsistent. The paper should identify which of the two is intended and correct the derivation accordingly; as presented, the contradiction is load-bearing because Section 4 claims to retrieve (19) from the new integral equation.","section":"Section 4, Eq. (20)"}],"minor_comments":[{"comment":"The section heading contains a repeated-text typo: 'S = µθS = µθS = µθ' should read 'S = µθ'.","section":"Section 4 heading"},{"comment":"The numerical verification of the time-dependent threshold should state the parameter values, the quadrature or discretization scheme, and the error tolerance; the current one-line statement is not reproducible.","section":"Section 3, numerical paragraph"},{"comment":"The phrase 'after some straightforward simplifications' hides a nontrivial normalization step in which the kernel is rewritten so that the threshold appears symmetrically in the exponential and in the D_q argument. Showing this intermediate step would make the algebra checkable and might have revealed the normalization error.","section":"Section 3, between Eqs. (13) and (14)"}],"recommendation":"reject","confidential_remarks":"The central claim of the paper is not supported: Eq. (14) conflicts with the known closed form (19) at the explicitly treated point q=1, x=S=μθ, producing an extra factor √2. The reader's specific objection about the exponential prefactor does not survive a careful rewrite of the kernels, but the √2 discrepancy is decisive and is not a local typo: it reflects an inconsistent normalization between the inverse Laplace transform input and the claimed Volterra equation. The time-dependent generalization is asserted without reproducible evidence. I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the paper's central equation (14) does not follow from the inversion formula (13). There is an algebraic error in the exponential terms, so the claimed new Volterra integral equation for the Ornstein–Uhlenbeck first passage density is not established. I would not send this to referees as is.\n\nWhat is genuinely new: the paper attempts to combine the author's earlier inverse Laplace transforms for products of parabolic cylinder functions with the convolution theorem to produce an integral equation whose kernel contains a parabolic cylinder function, with the Fortet equation as the q=0 special case. That is a reasonable idea, and if the algebra had worked out, it would be a useful addition to the numerical toolkit. The literature review is adequate, and the retrieval of the known closed form for S = μθ is correct.\n\nThe soft spot is fatal. In going from (13) to (14), the exponential term in the kernel changes from exp(−((μθ−S)+(x−μθ)e^{−s/θ})²/(2σ²θ(1−e^{−2s/θ}))) to exp(−((x−μθ)−(S−μθ)e^{−s/θ})²/(2σ²θ(1−e^{−2s/θ}))). These differ by a factor exp(((x−μθ)²−(S−μθ)²)/(2σ²θ)), which is independent of s but is not 1 for general x > S. The same missing factor appears on the right-hand side. So equation (14) is false for generic parameters, and equation (17) inherits the mistake. The claimed numerical verification in Section 3 is also not reproducible—no code, no parameter values—and cannot override an explicit algebraic contradiction. The special case S = μθ, where the missing factor equals 1, is retrieved correctly, but that does not test the general equation.\n\nThe paper is not a waste—the idea of using the author's Laplace inversion identities to construct a regular-kernel Volterra equation is worth pursuing, and the q=0 reduction to Fortet is a nice consistency check. But as it stands, the main result is incorrect. The author needs to fix the exponential factors and re-check the convolution step.\n\nWho this is for: someone working on numerical methods for first passage times of diffusions might find the approach interesting once corrected, but not in its current form. Recommendation: desk reject, with an explanation of the algebraic error; if the author fixes it and provides reproducible numerics, a fresh submission would be worth a referee.","headline":"The claimed new Volterra integral equation for OU first passage densities is invalid: equation (14) does not follow from (13) because the exponential terms are algebraically inconsistent, so the central result fails.","tokens_in":10221,"tokens_out":5541,"would_cite":false,"duration_ms":43052,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["33C15","44A10","44A35","45D05","60J35","60J70"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper derives a Volterra integral equation for the first-passage-time density of the Ornstein–Uhlenbeck process, with a parabolic-cylinder-function kernel, that reduces to the Fortet renewal equation at q=0 and covers time-dependent…","keywords":["first passage time","Ornstein–Uhlenbeck process","parabolic cylinder function","Volterra integral equation","Laplace transform","convolution theorem","Fortet renewal equation","time-dependent threshold"],"falsifier":"Evaluate the ratio of the exponential factor inside the inversion integral in (13) to the exponential kernel of (14) at generic parameter values (for instance x0=0, S=1, x=2, θ=1, σ=1, t=1, τ=0.5); if the ratio is not identically 1, the simplification claimed between those equations is false. Equivalently, compute both sides of (17) numerically for the exponential-threshold density (16) at q=1 and a non-degenerate parameter set and compare the discrepancy.","tokens_in":9205,"feed_emoji":"⏱","tokens_out":10079,"duration_ms":81420,"temperature":0.7,"pith_summary":"The paper aims to establish a new Volterra integral equation for the first-passage-time density of the Ornstein–Uhlenbeck process, with a kernel written in terms of a parabolic cylinder function. Unlike the standard Fortet renewal equation, whose kernel is weakly singular, the proposed kernel is claimed to be regular when the cylinder-function order satisfies q ≤ −1. The equation is asserted to hold for both constant and time-dependent thresholds, and the Fortet renewal equation emerges as the special case q = 0. If true, the result would give practitioners a new numerical route to first-passage-time probabilities in a widely used diffusion model.","feed_headline":"Parabolic-cylinder kernel gives a new first-passage equation","feed_subtitle":"Ornstein–Uhlenbeck first-passage densities via a regular Volterra kernel, with Fortet as q = 0.","key_machinery":"The mechanism is a Laplace-inversion identity for products of two parabolic cylinder functions (equation (9)), combined with the convolution theorem and the time-scaling property of the Laplace transform. The first-passage Laplace transform (a ratio of two parabolic cylinders) is multiplied by a chosen parabolic-cylinder function whose inversion is known; rearranging the resulting convolution yields the Volterra equation (14)/(17).","core_discovery":"The central claim is that equations (14) and (17) form a Volterra integral equation of the first kind for the first-passage-time density g(S(t), t | x0) of the Ornstein–Uhlenbeck process. The kernel is built from exponentials and a parabolic cylinder function D_q, with the order q available as a free parameter. For q = 0 the equation reduces exactly to the Fortet renewal equation. The paper also claims that the kernel is regular for q ≤ −1 at x = S(t), and that for x = S(t), q = 1 the equation is of the second kind and directly yields the known closed-form first-passage density for the constant threshold S = µθ, avoiding the usual transformation argument.","pith_inferences":["If the algebraic step between (13) and (14) does not hold, the integral equation itself could still be true; a direct proof by differentiation or renewal arguments, independent of Laplace inversion, would settle whether equation (17) stands.","The free parameter q suggests a family of integral equations; different choices of q might be equivalent by direct transformation of the transition density, which would give a way to tune the kernel's singularity profile for numerical work.","A numerical benchmark comparing the claimed regular kernel (q ≤ −1) against existing regularized forms of the Fortet equation on the exponential-threshold example would reveal whether the regularity translates into better accuracy or conditioning."],"forward_implications":["For q ≤ −1, the new equation's kernel is regular at x = S(t), so numerical evaluation may avoid the weakly singular kernel of the Fortet renewal equation.","At q = 0 the equation collapses to the classical Fortet renewal equation, making the Fortet equation a member of a one-parameter family.","For x = S = µθ and q = 1, the equation reduces to a Volterra equation of the second kind whose solution reproduces the closed-form density (19) without the usual transformation argument.","The same integral equation applies to time-dependent thresholds; the paper reports numerical agreement for the exponential threshold family (15) across values of q."],"supporting_citations":[{"why":"supplies the transition density and the Laplace transform of the first-passage density for the Ornstein–Uhlenbeck process that the convolution construction starts from.","marker":"[2]"},{"why":"introduces the renewal equation whose Ornstein–Uhlenbeck specialization is the q=0 limit of the claimed new equation.","marker":"[8]"},{"why":"provides the inverse Laplace transform identity for products of parabolic cylinder functions that underlies the kernel construction.","marker":"[14]"},{"why":"gives the product-of-two-parabolic-cylinder-functions inversion used to build the auxiliary Laplace transform pair.","marker":"[15]"},{"why":"supplies closed-form first-passage densities for exponential time-dependent thresholds, used as the test case for the time-dependent version.","marker":"[19]"},{"why":"states the explicit exponential threshold family and its first-passage density, used in the numerical verification of equation (17).","marker":"[20]"},{"why":"furnishes the special-function identities that reduce the kernel at q=0 to the Fortet equation and simplify the q=1 case.","marker":"[21]"}],"fun_headline_variants":["Volterra equation for Ornstein-Uhlenbeck first-passage via parabolic cylinder kernel","First-passage density via Volterra equation with parabolic cylinder kernel","Volterra first-passage equation reduces to Fortet at q=0","Parabolic cylinder kernel gives new Volterra equation for first-passage","Ornstein-Uhlenbeck first-passage density from new Volterra equation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation of equation (14) from (13) depends on an algebraic simplification of the exponential prefactors that must hold as an identity over all parameter values; if that equality is not exact, the stated Laplace-convolution argument does not produce the new integral equation.","fun_headline_variants_meta":{"raw":{"variants":["Volterra equation for Ornstein-Uhlenbeck first-passage via parabolic cylinder kernel","First-passage density via Volterra equation with parabolic cylinder kernel","Volterra first-passage equation reduces to Fortet at q=0","Parabolic cylinder kernel gives new Volterra equation for first-passage","Ornstein-Uhlenbeck first-passage density from new Volterra equation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002443,"raw_usage":{"total_tokens":9333,"prompt_tokens":838,"completion_tokens":8495,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":454,"completion_tokens_details":{"reasoning_tokens":8398}},"tokens_in":454,"tokens_out":8495,"duration_ms":55782,"temperature":1.0,"reasoning_tokens":8398,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:55:52.923595+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the ratio of the exponential factor inside the inversion integral in (13) to the exponential kernel of (14) at generic parameter values (for instance x0=0, S=1, x=2, θ=1, σ=1, t=1, τ=0.5); if the ratio is not identically 1, the simplification claimed between those equations is false. Equivalently, compute both sides of (17) numerically for the exponential-threshold density (16) at q=1 and a non-degenerate parameter set and compare the discrepancy.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the transition density and the Laplace transform of the first-passage density for the Ornstein–Uhlenbeck process that the convolution construction starts from."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces the renewal equation whose Ornstein–Uhlenbeck specialization is the q=0 limit of the claimed new equation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the inverse Laplace transform identity for products of parabolic cylinder functions that underlies the kernel construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the product-of-two-parabolic-cylinder-functions inversion used to build the auxiliary Laplace transform pair."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies closed-form first-passage densities for exponential time-dependent thresholds, used as the test case for the time-dependent version."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"states the explicit exponential threshold family and its first-passage density, used in the numerical verification of equation (17)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"furnishes the special-function identities that reduce the kernel at q=0 to the Fortet equation and simplify the q=1 case."}],"review_version":1}