{"id":"1137bc09-20d9-4367-aec6-f73bcf7b4110","arxiv_id":"2510.22416","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For affine-drift stochastic Volterra equations, the time-homogeneous Markov property forces the Volterra kernel to be exponential, K(t)=c e^{-λt}; a small-time CLT extends the failure to Hölder coefficients.","lead":"The paper proves that stochastic Volterra equations—unless their kernel is exponential—cannot be time-homogeneous Markov processes; the proof is rigorous for affine drifts and, via a small-time CLT, for Hölder-continuous coefficients. This gives rough volatility models and numerical schemes a precise statement of when exact Markovian reduction is impossible.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.3's resolvent-based second-moment formula (2.18) has a sign error; the proof of the zero-drift linear-diffusion characterization is invalid as written.","rationale":"I read the paper in good faith. The central contribution—failure of the time-homogeneous Markov property for SVEs—is supported by three affine theorems and a Hölder-coefficient theorem. Theorems 2.1 and 2.2 appear mathematically sound, and Theorem 4.2 is a careful reduction contingent on the external small-time CLT [18], exactly as the reader noted. However, Theorem 2.3 contains a concrete internal sign error in its second-moment resolvent computation. This is not a matter of an unproved external result or a weak assumption; it is a verifiable algebraic mistake in a key proof. Because Theorem 2.3 is one of the three results that establish the 'only exponential kernels' characterization for affine-drift SVEs, the paper as currently written does not fully support that claim for the zero-drift linear-diffusion case. The error seems correctable—replacing the minus sign with a plus in (2.18) may still lead to the exponential conclusion—but the proof must be fixed before acceptance. I recommend CONDITIONAL acceptance rather than unconditional ACCEPT. The reader's identified risk (reliance on [18]) is also legitimate, but my concern is distinct and more immediate.","tokens_in":18949,"tokens_out":28928,"duration_ms":248616,"concrete_test":"Re-derive Theorem 2.3 for the exactly solvable case λ=0, σ0=1, K(t)=1, x=1. Equation (2.18) evaluates to 2−e^T, while the true second moment of X_T=1+∫_0^T X_s dB_s is e^T, exposing the sign error. Then correct (2.18) to m(T)=x^2 e^{-2λT}+x^2∫_0^T e^{-2λ(T-s)}R_K(s) ds, recompute (2.19)–(2.22), and verify whether the derivation still forces K(t)=c e^{-λt}. If the corrected equations no longer imply exponential K, the theorem fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 2.3 is a pillar of the affine characterization, covering the zero-drift linear-diffusion case σ(x)=σ0x. Its proof rests on the second-moment representation (2.18), but that formula is wrong. Equation (2.17) reads m(T)=x^2 e^{-2λT}+σ0^2 ∫_0^T K(T-s)^2 m(s) ds. With K:=−σ0^2K^2, the equation is m=f−K*m. The resolvent R_K of K (in the sense of [21, Thm 2.3.5]) is defined by R_K=−K−K*R_K, and the solution to m=f−K*m is m=f+R_K*f, not f−R_K*f. Testing the elementary case λ=0, σ0=1, K≡1, x=1 gives the true second moment e^T, whereas (2.18) yields 2−e^T. Consequently, the subsequent equations (2.19)–(2.22), including the claim R_{K,λ}≤0, are based on a false premise. Theorem 2.3's proof therefore does not establish its conclusion as written. This is an internal algebraic error, independent of the external CLT [18] flagged by the reader.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies one-dimensional stochastic Volterra equations (SVEs) of the form X_t = x e^{-λt} + ∫_0^t K(t-s)b(X_s)ds + ∫_0^t K(t-s)σ(X_s)dB_s and asks when the associated family of laws can satisfy the time-homogeneous Markov property (1.3). The main positive results are necessary conditions on the Volterra kernel. Theorem 2.1 shows that for affine drift b(x)=b0+βx, under weak existence and a technical domain condition, Markovianity forces K(t)=ce^{-λt}. Theorems 2.2 and 2.3 treat the zero-drift cases σ(x)=σ0√x and σ(x)=σ0x, again concluding K(t)=ce^{-λt}. Section 3 gives a direct Gaussian computation proving that the fractional Riemann-Liouville process ∫_0^t (t-s)^{H-1/2}dB_s is not Markov for H≠1/2. Section 4, under Assumption 4.1 and using a small-time CLT from the authors' preprint [18], proves Theorem 4.2: if the limiting Gaussian process K*dB is not a time-homogeneous Markov process, then the SVE with Hölder coefficients cannot have the time-homogeneous Markov property. The paper is clearly written and the affine part is self-contained.","tokens_in":19261,"tokens_out":23019,"duration_ms":187432,"significance":"If the results hold, the paper provides the first rigorous and fairly general non-Markovity criterium for stochastic Volterra equations, a fact that is often taken for granted but rarely proved. The strength of the paper lies in the explicit moment computations: Theorems 2.1–2.3 reduce Markovianity to functional equations for resolvents and moments, and the conclusions are sharp for the examples (Volterra OU, Volterra square-root, Jacobi Volterra). Section 3's direct conditional-Gaussian proof is elegant and self-contained, and it gives an explicit quantitative signal (the asymptotic conditional mean of order τ^{1-2H}) rather than relying only on the Doob criterion. The assumptions of Section 4 are stated transparently, and the authors acknowledge the dependence on the companion preprint [18]. The main caveat is that Theorem 4.2 is only as reliable as that external CLT; this does not affect Theorems 2.1–2.3 or Section 3.","major_comments":[],"minor_comments":[{"comment":"The proof of Theorem 4.2 invokes [18, Theorem 2.2] as a black box, and Assumption 4.1 is essentially a restatement of its hypotheses. Since this is one of the two advertised approaches and the only one covering general Hölder coefficients, the manuscript would be easier to verify if the exact statement of the needed CLT were reproduced, or if Theorem 4.2 were explicitly labelled as conditional on [18]. This is a self-containedness/verifiability issue, not an internal inconsistency.","section":"Section 4 / Theorem 4.2"},{"comment":"I checked the potential sign issue in (2.18). With the resolvent convention used in [21] for the equation y=f−k*y, the resolvent r satisfies r=k−k*r and the solution is y=f−r*f. For k=−σ0^2K^2, formula (2.18) is correct, and the subsequent equations (2.19)–(2.22) are consistent. To avoid confusion, however, the authors should state this resolvent convention explicitly, since the notation R_K for the resolvent of −σ0^2K^2 collides with the earlier R_K for the affine drift problem.","section":"Theorem 2.3, Eq. (2.18)"},{"comment":"The abstract says that for affine drifts the Markov property only holds for exponential kernels. In the zero-drift case b≡0, which is affine, Theorems 2.2 and 2.3 cover only σ(x)=σ0√x and σ(x)=σ0x. The wording is slightly overbroad; suggest adding 'in the cases considered' or listing the covered diffusions.","section":"Abstract and Section 2"},{"comment":"The parametrization (β3,β2,β1)(τ)=(τ,τ^2,τ^3) is confusing because the displayed order is not increasing. Writing β1=τ^3, β2=τ^2, β3=τ with 0<τ<1 would make the ordering 0<β1<β2<β3 easier to read.","section":"Lemma 3.1"},{"comment":"The statement that the log-modulated fractional kernel satisfies (4.1) with γ*=H and arbitrary γ∈(0,H) is correct for small t, but the verification is only sketched via Karamata's theorem and a pointer to [18, Example 2.8]. Adding the short computation would make the example self-contained.","section":"Section 4, Example 4.4"}],"recommendation":"minor_revision","confidential_remarks":"The alleged sign error in Theorem 2.3 does not land: under the standard resolvent convention of [21], formula (2.18) and the subsequent argument are correct. The only substantive caveat is the reliance of Section 4 on the authors' own preprint [18]; if the journal is willing to accept citations to a posted preprint, the paper is acceptable after minor clarifications. The affine results and the Gaussian Section 3 are self-contained and constitute a solid contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the paper on stochastic Volterra equations and the Markov property. The headline: the affine characterization is likely correct, but Theorem 2.3 has a genuine proof error that needs fixing, and the paper leans on an unpublished CLT from the same group.\n\nWhat's actually new and good: Theorem 2.1 is the core result. It shows that for affine drift with b0 or β nonzero, the time-homogeneous Markov property forces the kernel to be exponential, using first-moment flow equations. The proof is transparent and self-contained. Theorem 2.2 handles the zero-drift square-root case and uses second moments instead. Section 3 gives a novel direct proof that the fractional Riemann-Liouville process is not Markov via conditional Gaussian means, though the fact itself was already in Bauer–Gerhold. Theorem 4.2 is a plausible reduction: if the small-time CLT from [18] holds, non-Markovianity of the Gaussian convolution implies non-Markovianity of the Hölder-coefficient SVE. That's a useful tool, but it depends on the authors' own preprint and Assumption 4.1 is stated, not proved, here.\n\nThe soft spot is real: in Theorem 2.3, equation (2.18) has the wrong sign. The second moment solves m = f − K*m with K = −σ0^2 K^2. The resolvent of K gives m = f + R_K*f, not f − R_K*f. A quick test with λ=0, σ0=1, K≡1, x=1 yields e^T as the true moment, while (2.18) gives 2−e^T. So the proof of Theorem 2.3 collapses as written. The conclusion may still be true — flipping the sign seems to convert the subsequent equations into the correct multiplicative form — but this is not a typo. It is a load-bearing algebraic error, and the authors need to redo that part.\n\nSecondary concerns: Section 3's non-Markovianity is not new, and Theorem 4.2 rests on the self-cited [18]. These are worth noting but not fatal.\n\nWho this is for: anyone working on Markovianity of Volterra processes and rough volatility. The moment approach is elegant and the question is folklore. The paper deserves refereeing, but the current version needs major revision, mainly fixing Theorem 2.3.","headline":"The affine characterization is appealing and likely correct, but Theorem 2.3's resolvent sign error makes the zero-drift linear case unproven as written.","tokens_in":19770,"tokens_out":8019,"would_cite":false,"duration_ms":62632,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G15","60G22","60H20","60J25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Stochastic Volterra equations are Markov only for exponential kernels","keywords":["stochastic Volterra equation","time-homogeneous Markov property","Volterra kernel","rough volatility","affine Volterra process","fractional Riemann-Liouville kernel","Gaussian process","small-time central limit theorem"],"falsifier":"Exhibit a single non-exponential continuous kernel K and coefficients satisfying the paper's assumptions for which the SVE (1.2) genuinely satisfies the time-homogeneous Markov property (1.3); Theorem 2.1 says no such example exists, so a valid example would refute the classification. Alternatively, find a kernel satisfying Assumption 4.1 for which K*dB is non-Markov but the SVE is Markov — Theorem 4.2 says this too is impossible.","tokens_in":18840,"feed_emoji":"⏳","tokens_out":7270,"duration_ms":63156,"temperature":0.7,"pith_summary":"The paper sets out to prove what practitioners have long assumed: stochastic Volterra equations, whose dynamics are shaped by a memory kernel, are not time-homogeneous Markov processes except in a narrow degenerate case. For equations with affine drift, it proves a sharp classification: if the solution family is time-homogeneous Markov, the Volterra kernel must be K(t)=c e^{-λt}, an exponential tied to the initial curve; for constant initial curves this means only the constant kernel, i.e. classical SDEs, qualifies. For Hölder-continuous coefficients, it proves a transfer theorem: under small-time power-law and CLT assumptions, non-Markovianity of the Gaussian process K*dB — which holds for fractional Riemann-Liouville kernels with H≠1/2 — implies non-Markovianity of the SVE. A sympathetic reader should care because rough volatility models and other path-dependent models are built on these equations, and their non-Markovianity is what makes them hard to simulate and hedge.","feed_headline":"Stochastic Volterra equations are Markov only for exponential kernels","feed_subtitle":"The paper shows path-dependent models from rough volatility to population dynamics are generally not memory-free.","key_machinery":"Two mechanisms carry the argument. First, the resolvent kernel E_K, the solution of E_K=K+β K*E_K, converts the affine SVE's mean into an explicit Volterra formula; Markovianity then forces the composed term E^λ_1 to be exponential, and the resolvent equation propagates this to K(t)=K(0)e^{-λt}. Second, the small-time central limit theorem rescales X_{t/n} by the inverse L^2 norm of K on [0,1/n]; if the SVE were Markov, the scaled process would converge to a Markov Gaussian process, so it suffices to test the Gaussian convolution K*dB. The test is the classical covariance condition for centered Gaussian processes, c(s,u)c(t,t)=c(s,t)c(t,u), and for the fractional Riemann-Liouville kernel the","core_discovery":"The central claim is Theorem 2.1: if a stochastic Volterra equation with affine drift b(x)=b_0+βx admits a continuous weak solution for every x in a closed domain D, and those laws satisfy the time-homogeneous Markov property (1.3), then the Volterra kernel must be K(t)=c e^{-λt} for all t, where λ is the exponential rate of the initial curve xe^{-λt}. Markovianity forces the first moment to satisfy a flow identity; comparing the direct moment formula with the Markov composition leads to the multiplicative relation E^λ_1(T)=E^λ_1(T-t)E^λ_1(t), whose only continuous solution is exponential, and the resolvent equation then propagates that exponential form to K. Theorem 4.2 extends this to Höld","pith_inferences":["A practical test emerges: before modelling with a candidate kernel, one can check the covariance condition for the Gaussian convolution K*dB; if that condition fails, the SVE cannot be Markov under the CLT assumptions.","The first-moment flow identity used in Theorem 2.1 gives a necessary consistency condition for any candidate transition kernel of an affine Volterra process, going beyond Markovianity itself.","The result suggests that Markovian approximation schemes for rough volatility are approximations in a strict sense: the true law is non-Markov, so any finite-dimensional Markov lift can only approximate, not reproduce, the path measure.","Extending the small-time CLT to conditional laws at positive times would turn Theorem 4.2 into a statement about time-inhomogeneous Markovianity, which is what many numerical methods actually assume."],"forward_implications":["For affine Volterra processes — Volterra Ornstein-Uhlenbeck, square-root (rough CIR), and Jacobi processes — the Markov property forces K(t)=c e^{-λt}; the rough CIR equation with a fractional kernel is therefore non-Markov.","With a constant initial curve (λ=0), the only Markovian SVEs among affine-drift equations are classical SDEs with constant kernel K≡c.","For Hölder coefficients, any kernel satisfying Assumption 4.1 whose scaled limit K*dB is non-Markov (e.g. fractional kernels H≠1/2, log-modulated fractional kernels) yields non-Markov SVEs, irrespective of the drift.","Exponential kernels are genuinely Markovian: they reduce the SVE to the semimartingale SDE dX_t=(K(0)b(X_t)-λX_t)dt+K(0)σ(X_t)dB_t.","The small-time CLT route is restricted to time-homogeneous Markovianity; proving failure of the time-inhomogeneous property would require a conditional CLT at arbitrary times, as the authors note."],"fun_headline_variants":["Volterra equations fail Markov property except exponential kernels","Only exponential kernels make Volterra models Markov","Volterra memory breaks Markov except for exponential kernels","No Markov property for Volterra equations without exponential kernels","Volterra Markov property holds only for exponential kernels"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The broad non-Markov theorem rests on an unproved small-time central limit theorem from a companion preprint; if that CLT or its exponent conditions fail, the reduction of SVEs to Gaussian processes collapses.","fun_headline_variants_meta":{"raw":{"variants":["Volterra equations fail Markov property except exponential kernels","Only exponential kernels make Volterra models Markov","Volterra memory breaks Markov except for exponential kernels","No Markov property for Volterra equations without exponential kernels","Volterra Markov property holds only for exponential kernels"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000769,"raw_usage":{"total_tokens":3209,"prompt_tokens":676,"completion_tokens":2533,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":420,"completion_tokens_details":{"reasoning_tokens":2462}},"tokens_in":420,"tokens_out":2533,"duration_ms":17361,"temperature":1.0,"reasoning_tokens":2462,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T08:06:10.526521+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a single non-exponential continuous kernel K and coefficients satisfying the paper's assumptions for which the SVE (1.2) genuinely satisfies the time-homogeneous Markov property (1.3); Theorem 2.1 says no such example exists, so a valid example would refute the classification. Alternatively, find a kernel satisfying Assumption 4.1 for which K*dB is non-Markov but the SVE is Markov — Theorem 4.2 says this too is impossible.","supporting_citations":[],"review_version":1}