{"id":"650001e1-8995-4efa-a98d-b69591b36374","arxiv_id":"2506.03421","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The normalized multilevel Euler error n^H(X^{mn}-X^n) converges stably to a Gaussian process for SVEs with fractional kernels, yielding a Lindeberg-Feller CLT for the MLMC estimator.","lead":"This paper proves central limit theorems for multilevel Monte Carlo estimators applied to stochastic Volterra equations with fractional kernels. The main result gives the asymptotic error distribution of the estimator, which is needed for confidence intervals and for tuning the number of samples per level.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.4 rests on unproved rate Hεn; for a simple linear SVE this rate is 2H, so for H<1/4 no α∈[1/2,1] exists and the claimed MLMC CLT does not cover the announced H-range.","rationale":"The reader's weakest assumption is exactly (Hεn), and I agree. I have made the concern concrete by exhibiting a simple linear zero-drift SVE where the weak error reduces to deterministic Volterra equations and has rate 2H. This does not invalidate Theorem 2.1, whose variance computation and H=1/2 limit match the known result of Ben Alaya and Kebaier; the damage is confined to the advertised generality of the MLMC CLT. Since the manuscript explicitly states (Hεn) as an assumption, a reader can treat Theorem 3.4 as conditional, but the abstract and introduction claim a CLT for MLMC estimators for H∈(0,1/2], which this example shows is unsupported. A revision should prove a sufficient weak-error bound, restrict the theorem to regimes where (Hεn) holds, or change the normalization and the limiting mean. The omitted Lemmas 2.4-2.8 are an additional proof gap, but even if those proofs were supplied, the (Hεn) issue would remain. The reader's conditional verdict is therefore retained.","tokens_in":54673,"tokens_out":16606,"duration_ms":216925,"concrete_test":"Set d=1, H=1/8, T=1, X0=1, b=0, σ(x)=x, f(x)=x^2. Solve the deterministic Volterra equations m(t)=1+∫_0^t K(t-s)^2 m(s)ds and m^n(t)=1+∫_0^t K(t-s)^2 m^n([ns]/n)ds to high accuracy for n=2^6,...,2^16, and compute ε_n=m^n(T)-m(T). Fit log|ε_n| against log n. If the fitted slope is approximately -2H=-1/4, or more generally lies above -1/2, then n^{1/2}ε_n diverges and assumption (Hεn) with α∈[1/2,1] fails for this example, showing that Theorem 3.4's normalization is not valid across the claimed H-range.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central MLMC result (Theorem 3.4) assumes (Hεn): lim n^α ε_n = C_f(T,α) with α∈[1/2,1], where ε_n = Ef(X^n_T)-Ef(X_T). Since Q_n telescopes to Ef(X^{m^L}_T), ε_n is exactly the single-level Euler weak error. The paper never proves such a rate; it only cites strong L^p rate n^{-H} (1.3), which is slower than n^{-1/2} when H<1/2. This is not merely an omitted proof. In the scalar case b=0, σ(x)=x, X0=1, f(x)=x^2, let m=E X^2 and m^n=E(X^n)^2. Then m=1+∫ K^2 m ds and m^n=1+∫ K^2 m^n([n·]/n) ds. The grid-cell increment m^n(s)-m^n([ns]/n) is O(n^{-2H}), so the forcing term in the error equation is O(n^{-2H}) and therefore ε_n=O(n^{-2H}). For H<1/4, 2H<1/2, so n^{1/2}ε_n diverges and no (Hεn) can hold. Hence Theorem 3.4, as stated for all H∈(0,1/2], does not apply to part of its own parameter range unless (Hεn) is proved for those cases. The omitted Lemmas 2.4-2.8 are a separate proof gap, but the rate assumption is decisive because it affects the statement, not only the proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves two limit theorems for the Euler discretization of the stochastic Volterra equation (1.1) with fractional kernel K(u)=u^{H-1/2}/Gamma(H+1/2). Theorem 2.1 states that the normalized two-level error U^{mn,n}=n^H(X^{mn}-X^n) converges stably in law in C_0^{H-epsilon} to the unique solution of a linear SVE driven by the original Brownian motion W and an independent Brownian motion B, with explicit volatility constant sqrt(g_m^H/(Gamma(2H+1)sin(pi H))). Theorem 3.4 states that, under the bias-rate assumption (Hεn), the multilevel Monte Carlo estimator Q_n satisfies n^alpha(Q_n-E f(X_T)) => N(C_f(T,alpha), sigma^2) with sigma^2 = Var-tilde(grad f(X_T) . U_T), for the sample sizes N_l specified in (3.16). The proof strategy is to expand the two-level error into low-order and martingale parts, compute all quadratic-variation limits in Section 4 and Appendices B-C, and then apply a Lindeberg-Feller argument to the MLMC decomposition.","tokens_in":1773,"tokens_out":1862,"duration_ms":202211,"significance":"If completed, Theorem 2.1 would be a valuable extension of the MLMC error CLT of Ben Alaya and Kebaier to singular Volterra kernels, and it would recover the Fukasawa-Ugai limit in the formal limit m -> infinity. The variance constants are derived analytically from the kernel and discretization rather than fitted, and the reduction to the classical H=1/2 case is a useful internal check. However, the paper is not yet in publishable form: the proof of Theorem 2.1 omits several load-bearing lemmas, and the proof of Theorem 3.4 has inconsistencies in the sample-size formulas and relies on an unproved, possibly false bias-rate assumption.","major_comments":[{"comment":"With the displayed choice N_l = n^{2alpha} m^{2(l-1)H} a_l / sum_{j=1}^L a_j, the level-0 term is not negligible. For l=0 this gives N_0 = n^{2alpha} m^{-2H} a_0 / A_L with A_L = sum_{j=1}^L a_j, so Var(n^alpha hat Q^1_n) = n^{2alpha} Var(f(X^1_T)) / N_0 = (A_L/a_0) m^{2H} Var(f(X^1_T)), which diverges. Hence the assertion n^alpha hat Q^1_n -> 0 in probability is false for the prescribed N_l. The separate formula for N_0 used in the proof is not the one in (3.16), and the deterministic bias epsilon_n cannot cancel the independent sample average hat Q^1_n. The statement must either choose N_0 with N_0 >> n^{2alpha} and redo the cost analysis, or include the level-0 variance in the asymptotic variance.","section":"§3, Eq. (3.16) and proof of Theorem 3.4"},{"comment":"The bias-rate assumption (Hεn) is load-bearing but is asserted rather than proved for singular kernels. In the scalar model b=0, sigma(x)=x, X_0=1, f(x)=x^2, the second-moment equations give m(t)=1+int_0^t K(s)^2 m(s)ds and m^n(t)=1+int_0^t K(s)^2 m^n([ns]/n)ds. The grid-cell error m^n(s)-m^n([ns]/n) is O(n^{-2H}), so epsilon_n = E f(X^n_T)-E f(X_T) is O(n^{-2H}). For H<1/4 this forces n^{1/2} epsilon_n -> infinity, so no alpha in [1/2,1] can satisfy (Hεn). Thus Theorem 3.4, as stated for all H in (0,1/2], has no content for part of its announced parameter range. The authors need to prove or cite a valid bias expansion for the relevant H-range, or restate Theorem 3.4 with an explicit rate restriction.","section":"§3, assumption (Hεn)"},{"comment":"The proof of Theorem 2.1 depends on Lemmas 2.4–2.8 for the convergence of the drift discretization remainders, tightness in C_0^{H-epsilon}, characterization of the limit, and strong uniqueness of the limiting equation. The manuscript states that these lemmas are analogous to those in [9] and omits their proofs. These steps are not routine consequences of the computations in Section 4, and the error process here is X^{mn}-X^n rather than X^n-X, so the cited arguments in [9] do not transfer verbatim. The proof of Theorem 2.1 is therefore incomplete as it stands.","section":"§2, Lemmas 2.4–2.8"},{"comment":"In the verification of the Lyapunov condition, the bound E|Z^{m^l,m^{l-1}}_{T,1}|^p <= K_p m^{-pl/2} is not the rate obtainable from Lemma A.6, which gives m^{-pHl} for H<1/2, and the displayed inequality sum_l n^p N_l^{-p/2} m^{-pl/2} <= tilde C_p (sum_l a_l)^{-p/2} sum_l a_l^{p/2} does not follow algebraically from (3.16). Since this is the step that establishes the Lindeberg condition, the proof of Theorem 3.4 needs a corrected calculation.","section":"§3, Lyapunov condition in Theorem 3.4"}],"minor_comments":[{"comment":"In the final displayed limit of Lemma 2.1(iii), the factor n^{4H} appears in front of the integral; the definition of the quadratic variation and all preceding displays require n^{2H}.","section":"Lemma 2.1(iii)"},{"comment":"The condition P(X_T /nelement D_f)=0 should read P(X_T notin D_f)=0.","section":"Theorem 3.4"},{"comment":"In Lemmas C.3 and C.5, 'cconverges' is a typo for 'converges'; additionally, in Lemma C.2 the process that is almost surely continuous should be H^{(m)}, not H.","section":"Appendix C.1"}],"recommendation":"major_revision","confidential_remarks":"I recommend major revision. The central ideas are promising, but the manuscript must repair the sample-size formula and the level-0 term in Theorem 3.4, supply the missing proofs of Lemmas 2.4–2.8, and reconcile the announced H-range with the bias-rate assumption (Hεn). The technical computations in Section 4 are extensive, but they do not by themselves resolve these load-bearing issues."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, Theorem 2.1 is a real new result: stable convergence of n^H(X^{mn}-X^n) for Volterra equations with fractional kernel K(u)=u^{H-1/2}/Γ(H+1/2), with a constant g_H^m that interpolates between the single-level limit of Fukasawa–Ugai and the H=1/2 MLMC limit of Ben Alaya–Kebaier. The variance computation in Lemma 2.1 is long but detailed, and the H=1/2 sanity check works. Appendix C's limit theorems for fractional integrals over two different grid sizes are genuinely new and non-obvious. This half of the paper deserves a serious referee.\n\nSecond, the MLMC part, Theorem 3.4, is not in shape as stated. It assumes the Euler weak error ε_n = Ef(X^n_T)-Ef(X_T) satisfies lim n^α ε_n = C_f(T,α) with α∈[1/2,1], and the paper never proves such a rate for singular kernels. The known strong rate is only n^{-H}, which is worse than n^{-1/2} when H<1/2. The stress-test concern is on target: in the scalar linear case b=0, σ(x)=x, f(x)=x^2, a direct calculation gives ε_n=O(n^{-2H}). For H<1/4 that is slower than n^{-1/2}, so no α∈[1/2,1] exists. The CLT as stated covers a parameter range where its hypothesis fails. Also, the proof's handling of the level-0 term is suspect: the substitution N0 = n^2(m-1)T / a0 Σaℓ does not match (3.16), and the claimed n\\hat Q^1_n → 0 is not justified as written.\n\nThe omitted Lemmas 2.4–2.8 (tightness, limit identification, uniqueness) are load-bearing for Theorem 2.1. Saying they are 'similar' to Fukasawa–Ugai is not enough, because the two-level error has a genuinely different correlation structure. These proofs need to be provided.\n\nWho this is for: researchers in computational probability working on MLMC for non-Markovian Volterra equations. The paper is worth reading for Theorem 2.1 and Appendix C, and it deserves peer review, but not acceptance in current form. The authors should either prove a weak error rate under explicit assumptions or restrict Theorem 3.4 to the range where (Hεn) can hold.","headline":"Genuinely new two-level error CLT for Volterra Euler schemes, but the MLMC CLT leans on an unproved and likely false bias-rate assumption.","tokens_in":55591,"tokens_out":3948,"would_cite":false,"duration_ms":41614,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F05","60H20","62F12","65C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For stochastic Volterra equations with fractional kernels, the multilevel Monte Carlo Euler estimator satisfies a central limit theorem whose limiting variance is identified explicitly and whose two-level error converges stably in law.","keywords":["Stochastic Volterra integral equations","Multilevel Monte Carlo","fractional kernels","stable convergence","central limit theorem","Euler-Maruyama scheme","Lindeberg-Feller condition"],"falsifier":"For the linear SVE with b=0, \\$\\sigma$=1 and H<1/2, the solution is X_t=\\int_0^t K(t-s)dW_s, a Gaussian process whose covariance is known; compute \\epsilon_n=E f(X^n_T)-E f(X_T) exactly or by very accurate simulation for f(x)=x and increasing n. If |\\epsilon_n| decays like $n^{{-H}}$ rather than $n^{{-\\alpha}}$ for some \\$\\alpha$\\ge 1/2, then the bias-rate assumption (H\\epsilon n) fails and the mean-shift term of Theorem 3.4 is not supported.","tokens_in":54416,"feed_emoji":"📊","tokens_out":9692,"duration_ms":96123,"temperature":0.7,"pith_summary":"Stochastic Volterra equations with fractional kernels K(u)=$u^{{H-1/2}}$/\\Gamma(H+1/2), H\\in(0,1/2], are used when a model needs memory or rough volatility, but their solutions are rarely explicit, so simulation relies on Euler schemes and multilevel Monte Carlo (MLMC). This paper proves that the normalized two-level Euler error n^H($X^{{mn}}$-X^n) converges stably in law to a linear stochastic Volterra equation with an explicit volatility constant, and that the MLMC estimator obeys a Lindeberg-Feller central limit theorem. If the theorems are correct, the heuristic practice of using MLMC variance formulas for such equations becomes a theorem, and the limiting distribution supplies the error constant needed for sample-size allocation and confidence intervals.","feed_headline":"Multilevel Euler errors obey a central limit theorem","feed_subtitle":"For fractional-kernel Volterra equations, the two-level error's limiting distribution is explicit, enabling MLMC inference.","key_machinery":"The carrying object is the two-level error process $U^{{mn,n}}$=n^H($X^{{mn}}$-X^n), built from Euler schemes with step sizes 1/(mn) and 1/n on the same Brownian path. The proof expands the coefficient differences by Taylor's formula, isolates a vanishing drift term from a martingale term, and computes the $L^{2}$ limit of the quadratic variation of the associated stochastic integrals. The fractional-kernel calculus is the mechanism that turns discrete kernel increments into the constant g^H_m=\\sum_{j=0}^{m-1}$j^{{2H}}$/$m^{{2H+1}}$ and the identity 2H\\int_0^\\infty|\\mu(r,1)|^2 dr+1=G\\,\\Gamma(2H)\\sin\\pi H, with G=\\Gamma(H+1/2)^2, yielding the explicit limiting volatility.","core_discovery":"The paper's central claims are two. Theorem 2.1: for H\\in(0,1/2], fixed integer m\\ge 2, and \\epsilon\\in(0,H), the process $U^{{mn,n}}$_t=n^H($X^{{mn}}$_t-X^n_t) converges stably in law in $C^{{H-\\epsilon}}$_0 to the unique solution U of the linear stochastic Volterra equation (2.10), driven by the original Brownian motion W and an independent Brownian motion B, with diffusion coefficient \\sqrt{g^H_m/(\\Gamma(2H+1)\\sin\\pi H)}, where g^H_m=\\sum_{j=0}^{m-1}$j^{{2H}}$/$m^{{2H+1}}$. Theorem 3.4: if f satisfies the stated growth and differentiability condition, P(X_T\\notin D_f)=0, and the Euler bias obeys n^\\$\\alpha$(E f(X^n_T)-E f(X_T))\\to C_f(T,\\$\\alpha$) for some \\$\\alpha$\\in[1/2,1], then with independent samples and sample sizes N_l=$n^{{2\\alpha}}$$m^{{-2H(l-1)}}$a_l/\\sum a_l, n^\\$\\alpha$(Q_n-E f(X_T))\\Rightarrow N(C_f(T,\\$\\alpha$),\\$sigma^{2}$), where \\$sigma^{2}$ is the variance of \\nabla f(X_T)\\cdot U_T.","pith_inferences":["The paper leaves open whether the bias-rate assumption (H\\epsilon n) actually holds for H<1/2; since the known Euler error bound is only n^{-H}, the mean-shift term C_f(T,\\alpha) is conditional and deserves a separate proof or disproof.","The explicit dependence of the limiting variance on g^H_m suggests that the level ratio m in the MLMC scheme can be optimized, and for small H the geometric decay m^{-2H\\ell} may change the optimal choice of m compared with the Brownian case.","Stable convergence with an independent Brownian motion B appearing in the limit hints that the multilevel error is asymptotically uncorrelated with the increments of W, which could be exploited for control variates or for constructing a Richardson-type extrapolation of the error."],"forward_implications":["For H=1/2, Theorem 2.1 reproduces the known MLMC Euler central limit theorem for classical Itô SDEs, and letting m tend to infinity recovers the single-level Euler error central limit theorems of Jacod-Protter and Fukasawa-Ugai.","The normalized two-level error converges stably in law, so the limiting process is defined on an extension of the original probability space and is conditionally Gaussian given the driving Brownian path.","The MLMC variance satisfies Var(Q_n)=O(\\sum_{\\ell=0}^L N_\\ell^{-1} m^{-2H\\ell}), giving a quantitative guide for choosing sample sizes per level.","Under the bias-rate assumption (H\\epsilon n), the central limit theorem provides the asymptotic normal distribution with explicit mean shift C_f(T,\\alpha) and variance \\sigma^2, so confidence intervals for E f(X_T) can be constructed."],"supporting_citations":[{"why":"supplies the stable-limit framework for the single-level Euler error of SVEs with fractional kernel, which the two-level result extends.","marker":"[9]"},{"why":"is the H=1/2 MLMC Euler central limit theorem that Theorem 2.1 recovers as a special case and Theorem 3.4 generalizes.","marker":"[4]"},{"why":"is the classical Euler error central limit theorem for SDEs that is recovered formally as m tends to infinity at H=1/2.","marker":"[16]"},{"why":"independently proved the single-level Euler error distribution for SVEs, supporting the limit equation (1.5).","marker":"[24]"},{"why":"provides related singular-kernel limit theorems for the Milstein scheme that the authors adapt to the two-level error.","marker":"[21]"},{"why":"supplies the representation of conditional Gaussian martingales and stable convergence used in Lemma 2.3.","marker":"[15]"},{"why":"contains the fractional-integral identity 2H\\int|\\mu(r,1)|^2 dr+1=G\\,\\Gamma(2H)\\sin\\pi H that evaluates the limiting volatility constant.","marker":"[23]"},{"why":"proves the n^{-H} Euler rate for SVEs with weakly singular kernels used to set the bias-rate context.","marker":"[20]"}],"fun_headline_variants":["CLT for multilevel Euler estimates of Volterra equations","Explicit limit law for MLMC Euler error in Volterra equations","Multilevel Monte Carlo Euler error obeys central limit theorem","Fractional Volterra equations: error distribution made explicit","Stable convergence for multilevel Euler in Volterra SDEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The MLMC central limit theorem assumes that the bias of the Euler estimator decays no slower than the square root of the step size; the paper does not prove this for singular kernels, and its own error bound gives only the slower decay rate $n^{{-H}}$, so the mean-shift constant in the limit is conditional on that unproved rate.","fun_headline_variants_meta":{"raw":{"variants":["CLT for multilevel Euler estimates of Volterra equations","Explicit limit law for MLMC Euler error in Volterra equations","Multilevel Monte Carlo Euler error obeys central limit theorem","Fractional Volterra equations: error distribution made explicit","Stable convergence for multilevel Euler in Volterra SDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000172,"raw_usage":{"total_tokens":1242,"prompt_tokens":879,"completion_tokens":363,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":279}},"tokens_in":495,"tokens_out":363,"duration_ms":3377,"temperature":1.0,"reasoning_tokens":279,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:05:21.718087+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the linear SVE with b=0, \\$\\sigma$=1 and H<1/2, the solution is X_t=\\int_0^t K(t-s)dW_s, a Gaussian process whose covariance is known; compute \\epsilon_n=E f(X^n_T)-E f(X_T) exactly or by very accurate simulation for f(x)=x and increasing n. If |\\epsilon_n| decays like $n^{{-H}}$ rather than $n^{{-\\alpha}}$ for some \\$\\alpha$\\ge 1/2, then the bias-rate assumption (H\\epsilon n) fails and the mean-shift term of Theorem 3.4 is not supported.","supporting_citations":[{"cited_title":"Fukasawa and T","cited_arxiv_id":null,"evidence_quote":"supplies the stable-limit framework for the single-level Euler error of SVEs with fractional kernel, which the two-level result extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"is the H=1/2 MLMC Euler central limit theorem that Theorem 2.1 recovers as a special case and Theorem 3.4 generalizes."},{"cited_title":"Jacod and P","cited_arxiv_id":null,"evidence_quote":"is the classical Euler error central limit theorem for SDEs that is recovered formally as m tends to infinity at H=1/2."},{"cited_title":"Nualart and B","cited_arxiv_id":null,"evidence_quote":"independently proved the single-level Euler error distribution for SVEs, supporting the limit equation (1.5)."},{"cited_title":"Limit error distributions of Milstein scheme for stochastic Volterra equations with singular kernels","cited_arxiv_id":"2412.11126","evidence_quote":"provides related singular-kernel limit theorems for the Milstein scheme that the authors adapt to the two-level error."},{"cited_title":"Jacod, On continuous conditional Gaussian martingales and stable convergence in law , Séminaire de Probabil- ités, XXXI, Lecture Notes in Math., vol","cited_arxiv_id":null,"evidence_quote":"supplies the representation of conditional Gaussian martingales and stable convergence used in Lemma 2.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"contains the fractional-integral identity 2H\\int|\\mu(r,1)|^2 dr+1=G\\,\\Gamma(2H)\\sin\\pi H that evaluates the limiting volatility constant."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"proves the n^{-H} Euler rate for SVEs with weakly singular kernels used to set the bias-rate context."}],"review_version":1}