{"id":"5229451e-01b1-4bbf-9c0a-aeeac055995f","arxiv_id":"2507.11429","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For SDEs with Hölder drift driven by symmetric α-stable noise, α∈(1,2), the randomised Euler-Maruyama scheme has L^p strong order 1/2 + β ∧ (η/α) ∧ 1/2 − ε, above the standard EM order.","lead":"This paper analyzes a faster way to simulate stochastic differential equations whose noise has heavy-tailed jumps and whose drift is only Hölder continuous. It proves the randomised Euler-Maruyama scheme reaches strong order 1/2 + (β ∧ (η/α) ∧ 1/2) − ε, beating standard Euler-Maruyama when the drift is rough in time.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"In Proposition 3.6 the δA^1 estimate discards the spatial-Hölder factor of g2 in J2/J4, giving an h-power that can violate the shifted SSL (Theorem 2.7) hypotheses; the proof needs a repair even if the rate is true.","rationale":"The reader's CONDITIONAL verdict is appropriate, and I agree that the central claim is plausible and well supported by the numerical section. However, the most specific obstacle to the proof is not only an unverified external lemma but an algebraic step inside Proposition 3.6 that drops a factor needed for the stochastic sewing argument. The reader's weakest_assumption mentions the external lemmas and measurability in the same propositions; my concern is adjacent but more precise, hence 'partial'. The proposed check is a one-line re-derivation that settles whether the bound (26) follows from the stated hypotheses. If the check fails, a fix exists for the particular g2 used in Theorem 2.9, so the verdict should remain CONDITIONAL rather than moving to ACCEPT or REJECT.","tokens_in":23650,"tokens_out":24980,"duration_ms":272606,"concrete_test":"Recompute the J2 (and J4) term in Proposition 3.6 keeping the spatial-Hölder bound |g2(r,X_{s1})-g2(r,X_{s2})| ≤ C|X_{s1}-X_{s2}|^ϖ and using Lemma 3.1 to insert the extra |t-s|^{ϖ/α-ε}. Verify whether the resulting δA^1 estimate has h-exponent strictly greater than 1, so that the hypotheses of Theorem 2.7, condition (16), hold. If it does, the proof is repairable; if not, determine the additional condition on (α,β,η,ϖ) and amend Proposition 3.6 or Theorem 2.9 accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive step is the estimate of E_{s1}δA^1(s,u,t) in Proposition 3.6. The J2 term is bounded by replacing g2(r,X_{s1})-g2(r,X_{s2}) with 2∥g2∥∞, so it is estimated as C∫ |r-s2|^{(η-1)/α} ∥E_{s2}[φ_r-φ_{κn(r)}]∥ dr ≤ C n^{-1}|t-s|. Retaining the Hölder factor gives |g2(X_{s1})-g2(X_{s2})| ≤ C∥g2∥_{C^{0,ϖ}} |X_{s1}-X_{s2}|^ϖ, and Lemma 3.1 yields an extra |t-s|^{ϖ/α-ε}; without it the integral is C n^{-1}|t-s|^{1+(η-1)/α}, not n^{-1}|t-s| when η<1. Condition 2η+α>2 only gives 1+(η-1)/α>1/2, not >1; for example η=0.5, α=1.5 gives exponent 2/3. Theorem 2.7 requires in condition (16) h-powers strictly greater than 1, so a n^{-1}h^{2/3} term cannot be absorbed into Γ2h^{1+ε2}+Γ3h^{1+ε3} with finite Γ's independent of h. Therefore the conditional shifted SSL is not applicable to A^1 as written, and the bound (26) does not follow from the proof given. The gap seems repairable in the main application because g2=∇u has ϖ=(η/2+α-1)∧1, making η+ϖ>1 for the relevant parameters, but Proposition 3.6 as stated and proved requires modification.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyses the randomised Euler-Maruyama method (6) for additive time-inhomogeneous SDEs (1) driven by a symmetric alpha-stable Levy process with alpha in (1,2) and drift coefficient b in C^{beta,eta}_b. The main result, Theorem 2.9, claims that for all p>=1 and epsilon in (0,1/2), E[sup_{0<=t<=1}|X_t - X_t^{(n)}|^p] <= C n^{-(1/2+gamma-epsilon)p}, where gamma = beta ∧ (eta/alpha) ∧ (1/2), under the conditions 2eta+alpha>2 and (beta+1)alpha+eta>2. The proof strategy combines a deterministic-vs-noise decomposition of the numerical solution, stochastic sewing lemmas (standard and conditional shifted), external estimates from Butkovsky et al. [10], and a Zvonkin-type transformation; the paper also contains numerical experiments for several irregular drift functions.","tokens_in":24082,"tokens_out":11573,"duration_ms":136525,"significance":"If the proof were complete, the result would be a significant extension of the randomised EM analysis from Gaussian noise to stable Levy noise: it improves on the standard EM order, which is limited by the time-regularity beta of the drift, and matches the near-optimal rate known for time-homogeneous stable SDEs. The paper also provides useful numerical evidence. The main concern is that the proof of the decisive intermediate bounds relies on unverified transfers of external lemmas and contains a specific error in the estimate of the conditional sewing increments in Proposition 3.6; because these bounds feed directly into Theorem 2.9, the central claim is plausible but not established as written.","major_comments":[{"comment":"The estimate of Es-(t-s) δA^1(s,u,t) is not sufficient for the application of the conditional shifted stochastic sewing lemma (Theorem 2.7). In the J2 and J4 terms the proof replaces |g2(r,X_{s1}) - g2(r,X_{s2})| and |g2(r,X_{s1}) - g2(r,X_{s3})| by 2||g2||∞, which yields contributions of order C n^{-1} |t-s|^{1+(eta-1)/alpha} when combined with the bounds on Es2[phi_r - phi_{kappa_n(r)}]. The displayed final bound writes the second term as C n^{-1} |t-s|, but for eta<1 we have 1+(eta-1)/alpha < 1, so this replacement goes in the wrong direction. More importantly, even the stated C n^{-1}|t-s| does not have the form Gamma h^{1+epsilon} required in condition (16) of Theorem 2.7, since n^{-1}h cannot be absorbed into a constant independent of h without losing the epsilon. Thus the conditional shifted SSL is not applicable to A^1 as written, and the bound (26) is not established by the given proof. The main application may be repairable by retaining the spatial Holder factor of g2 and using the specific value ϖ=(eta/2+alpha-1)∧1 from Theorem 2.8, but Proposition 3.6 as stated and proved requires modification.","section":"Proposition 3.5"},{"comment":"The transfer of [10, Lemma 4.4] and [10, Lemma 4.7] to the present randomised scheme is asserted rather than verified. The text says that condition (19) 'ensure[s] the applicability of [10, Lemma 4.4]' with parameter choices theta=eta, tau=1+beta∧(eta/alpha)-epsilon, epsilon0=1, gamma=1/2, and it invokes [10, Lemma 4.7] for the I2-type bound after claiming that phi^{(n)}_{kappa_n(t)} is F_{(kappa_n(t)-1/n)∨1}-measurable. However, definition (20) shows that phi^{(n)}_t depends on tau_{floor(nt)+1}, so this measurability assertion is not immediate and needs a proof. The hypotheses of the two external lemmas (filtration structure, adaptedness, conditional moment bounds, and applicability to processes defined with the joint law of (L,tau)) are not checked. Since estimate (23) is used in Proposition 3.7 and hence in the proof of Theorem 2.9, this gap is load-bearing; the authors should either verify the hypotheses explicitly or provide self-contained proofs.","section":"Proposition 3.7"},{"comment":"The passage from the conditional estimate E_{k/n}|A^1_t - A^1_s| <= C n^{-(1/2+gamma-epsilon)} to the L^p sup-norm bound invokes the Weighted John-Nirenberg inequality [10, Proposition 3.2], but the required hypotheses of that result (for example, that the estimate holds for the full dyadic family of intervals with constants independent of the interval, and that the filtration and conditioning sigma-algebras are the ones used there) are not verified. This sup-norm bound is what enters the Gronwall argument in (37) through (35)-(36), so this is another point where the route from the local estimates (23) and (26) to the final theorem needs to be completed explicitly.","section":"Section 4"},{"comment":"In the estimation of Theta4, the small-jump bound uses the inequality |H(t,z)| <= C ||u||_{C^{0,alpha+eta}_b} |X_{t-} - X_{t-}^{(n)}| |z|^{alpha+eta-1} for |z|<=1, attributed to [25, Lemma 4.1]. The integrability of |z|^{2(alpha+eta-1)} against the Levy measure does use the condition 2eta+alpha>2, and this works out, but the presentation is marred by the apparent interchange of the labels Theta_{4,1} and Theta_{4,2} for the large- and small-jump components. More substantively, the line following (31) applies to the |z|>1 term but is labelled Theta_{4,2}; the authors should correct the labels and make explicit which bound is used for each jump-size regime.","section":"Theorem 2.9"}],"minor_comments":[{"comment":"The phrase 'symmetric α-table process' in the abstract should read 'symmetric α-stable process'.","section":"Abstract and Introduction"},{"comment":"The norm notation in the statement and proof of Proposition 3.6 is inconsistent: ||g1||_{C^{α,β}_b} should be ||g1||_{C^{β,η}_b}, and C([0,1]; C^{0,ϖ}_b) should be written consistently as C([0,1]; C^{ϖ}_b).","section":"Proposition 3.6"},{"comment":"The proof of (22) defines the auxiliary process A_{s,t} with g2(kappa_n(r), X^{(n)}_{kappa_n(r)}), whereas the statement of (22) contains g2(r, X^{(n)}_r); this mismatch between the sewing increment and the target integral should be reconciled explicitly.","section":"Lemma 3.2"},{"comment":"The displayed pointwise bound b(t,X_t) <= ||b||_{C^{β,η}_b} (and its analogue for the numerical solution) should use absolute values, namely |b(t,X_t)| <= ||b||_{C^{β,η}_b}, since the drift is vector-valued and can be negative in each component.","section":"Remark 2.10"},{"comment":"There are several typographical errors throughout, including 'Mckean-Valsov', 'equiped', and 'stbale'; a careful proofreading pass is recommended.","section":"Global"}],"recommendation":"major_revision","confidential_remarks":"The central claim is plausible and within the journal's scope, but the proof as written has a specific error in Proposition 3.6 and an unverified transfer of external lemmas in Proposition 3.5; both are load-bearing for Theorem 2.9. If the authors can repair the δA^1 estimate (for instance by using the Hölder regularity of g2 and the special value of ϖ in the main application) and verify the hypotheses of [10, Lemmas 4.4 and 4.7] and of the John-Nirenberg inequality, the paper could become publishable. I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know: this paper gets the right answer but has a real hole in one of the two central estimates. The main result—randomised Euler–Maruyama for additive time-inhomogeneous SDEs driven by symmetric α-stable noise, α∈(1,2), with Hölder drift in space and time—is new and important. The achieved rate 1/2 + β∧(η/α)∧1/2 − ε is exactly what one would hope for: it extends the Gaussian result from [6], matches the sharp time-homogeneous stable-EM rate from [10], and improves on standard EM, which cannot exceed β. The proof architecture is sensible: decompose the error via the Zvonkin transform, use stochastic sewing, and control the martingale terms with the conditional shifted sewing lemma. The numerical experiments support the claimed improvement, and the authors are honest about which examples fall outside the assumptions. Credit is due for tackling the right problem and getting the rate right.\n\nBut there is a substantive gap in Proposition 3.6, exactly as the stress-test note identifies. In estimating δA^1, the J2 term is bounded by replacing the spatial Hölder factor of g2 with 2∥g2∥∞, losing the |X_{s1}−X_{s2}|^ϖ factor. The resulting integral is C n^{-1}|t−s|^{1+(η−1)/α}, and the assumption 2η+α>2 only gives exponent >1/2, not >1. Theorem 2.7 requires h-powers strictly greater than 1 in condition (16), so this term cannot be absorbed into the shifted-SSL error with finite constants. The bound (26) therefore does not follow as written for general g2∈C^{0,ϖ}. This is likely repairable—in the application g2=∇u has better spatial regularity—but Proposition 3.6 overclaims as stated. There is also the related concern that Propositions 3.5 and 3.6 lean on external lemmas [10, Lemma 4.4 and 4.7] without fully verifying hypotheses in the randomised joint-filtration setting. Plus smaller issues: Lemma 3.3 has an unexplained 1/p exponent, the manuscript has several typos (e.g., “α-stbale”), and the numerical section mentions a GitHub repository without giving the URL or error bars.\n\nWho is this for? Numerical probabilists studying strong approximation of SDEs with irregular coefficients, especially with Lévy noise. It is a serious paper with a likely-correct central theorem, but it is not fully proven as written. A thoughtful referee could help the authors close the gap. Given the importance of the result and the repair path, this deserves peer review rather than desk rejection. My advice: send it out, but ask the referees to focus on Proposition 3.6 and the transfer of [10]'s lemmas. If that gets fixed, this will be a solid contribution.","headline":"A significant and plausible extension of randomised EM to stable-driven SDEs, but one of the two key estimates has a genuine gap that needs repair.","tokens_in":24606,"tokens_out":2396,"would_cite":true,"duration_ms":30991,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65C30","65C05","60G51","60H10","60H35","60L90"],"pacs":[],"model":"deepseek-v4-flash","headline":"For SDEs with Hölder drift driven by symmetric α-stable Lévy noise, randomised Euler–Maruyama attains strong order 1/2 + γ − ε with γ = β∧(η/α)∧(1/2), above the standard EM order β.","keywords":["randomised Euler–Maruyama","α-stable Lévy process","Hölder continuous drift","strong order of convergence","stochastic sewing lemma","irregular drift","heavy-tailed noise","time-inhomogeneous SDE"],"falsifier":"Inspect the proof's hinge: verify directly that Lemma 4.4 and Lemma 4.7 of reference [10] apply to the randomised conditional process. Concretely, compute $\\|E_s[|\\phi^{(n)}_t - E_s\\phi^{(n)}_t|^2]\\|_{L^2(\\Omega)}$ from (20) and check whether the exponent $1+\\beta\\wedge(\\eta/\\alpha)-\\varepsilon$ and the stated $\\mathcal{F}_{(\\kappa_n(t)-1/n)\\vee 1}$-measurability of $\\phi^{(n)}_{\\kappa_n(t)}$ hold on the joint filtration; a single parameter triple $(\\alpha,\\beta,\\eta)$ satisfying (19) where either check fails refutes the proof of Theorem 2.9.","tokens_in":23412,"feed_emoji":"🎲","tokens_out":12466,"duration_ms":136912,"temperature":0.7,"pith_summary":"This paper proves a strong convergence rate for the randomised Euler–Maruyama scheme applied to additive SDEs driven by symmetric α-stable Lévy processes, α∈(1,2), when the drift is β-Hölder in time and bounded η-Hölder in space. The scheme evaluates the drift at a uniformly random point inside each time cell, and the paper shows the L^p error satisfies E[sup_{0≤t≤1}|X_t−$X^{{(n)}}$_t|^p] ≤ C $n^{{−(1/2+γ−ε)p}}$ with γ = β∧(η/α)∧(1/2). That order is higher than standard Euler–Maruyama, whose order cannot exceed β in this time-inhomogeneous setting, and it matches the near-optimal rate known for time-homogeneous stable SDEs and for the Gaussian case α=2. A sympathetic reader would care because α-stable noise models heavy-tailed, jumpy randomness, and the result shows that one cheap extra random draw per step buys a genuine order improvement without changing the Euler structure.","feed_headline":"Randomised Euler-Maruyama outruns EM for stable SDEs with Hölder drift","feed_subtitle":"Sampling the drift at a random point in each interval gains half an order of strong convergence over EM for heavy-tailed noise.","key_machinery":"The machinery has four parts. (i) The randomised evaluation map $\\kappa^\\tau_n$ converts time irregularity into a conditional expectation over a fresh uniform variable, which is what removes the $\\beta$ ceiling. (ii) A Zvonkin-type change of variables via the resolvent equation $\\partial_t u-\\lambda u+\\mathcal{L}u+b\\nabla u+b=0$ (Theorem 2.8) converts the drift error into a term controlled by $\\lambda\\|u(t,X_t)-u(t,X^{(n)}_t)\\|$ plus bang-bang terms. (iii) The stochastic sewing lemma (Theorem 2.6) and its conditional shifted version (Theorem 2.7) turn local conditional-moment bounds (Lemmas 3.2–3.4) into sup-norm $L^p$ bounds. (iv) Two technical lemmas from reference [10] (Lemma 4.4 and Lemma 4.7) are invoked in Propositions 3.5–3.6 to bound the two pieces $I_1$ and $I_2$ of the drift error; the conditions $2\\eta+\\alpha>2$ and $(\\beta+1)\\alpha+\\eta>2$ are exactly what their parameter choices require.","core_discovery":"The central claim is Theorem 2.9. For a $d$-dimensional symmetric $\\alpha$-stable Lévy process $L$ with $\\alpha\\in(1,2)$ and a drift $b\\in C^{\\beta,\\eta}_b$ satisfying $2\\eta+\\alpha>2$ and $(\\beta+1)\\alpha+\\eta>2$, the continuous randomised Euler–Maruyama approximation (6) satisfies\n$$\nE\\Big[\\sup_{0\\le t\\le 1}\\big|X_t-$X^{{(n)}}$_t\\big|^p\\Big]\\le C $n^{{-(1/2+\\gamma-\\varepsilon)p}}$\n$$\nfor every $p\\ge 1$ and $\\varepsilon\\in(0,1/2)$, where $\\gamma=\\beta\\wedge(\\eta/\\alpha)\\wedge(1/2)$. The scheme evaluates the drift at $\\kappa^\\tau_n(s)=\\lfloor ns\\rfloor/n+\\tau_{\\lfloor ns\\rfloor+1}/n$ with i.i.d. uniform $\\tau_i$, so the time-irregular part of the drift is averaged out rather than sampled only at grid points. This order exceeds the standard EM ceiling $\\beta$ and, in the $\\alpha=2$ limit, reduces to the known Gaussian randomised EM result.","pith_inferences":["Editorial inference: the same one-random-point-per-cell recipe should carry over to multiplicative stable noise or to mean-field SDEs, as long as analogous conditional-moment and semigroup gradient estimates hold.","Editorial inference: the numerical examples with discontinuous or non-Lipschitz drifts suggest the gain is not tied to the Hölder hypotheses; a natural test is to prove a version for bounded measurable drifts and check whether the half-order baseline persists.","Editorial inference: making the proof independent of the two external lemmas from reference [10] by proving direct conditional-moment bounds would clarify the optimality of the rate and likely extend it to α≤1 or to the boundary case β=0."],"forward_implications":["For every admissible (α,β,η), the scheme's sup-in-time L^p error is bounded by C n^{−(1/2+γ−ε)p} for all p≥1, so the gain over standard EM is at least a half order whenever γ > β − 1/2.","Since one extra uniform draw per step is negligible compared with computing the Lévy increments, the order improvement is essentially free in computational cost.","The order formula saturates: once β≥1/2 and η/α≥1/2, further Hölder smoothness does not improve the strong convergence rate beyond 1−ε.","The proof also extends the PDE-based EM error analysis from truncated to full symmetric α-stable processes, and the numerical experiments show lower errors for the randomised scheme across the whole discretization range."],"supporting_citations":[{"why":"Supplies the conditional shifted stochastic sewing lemma, the semigroup estimates in Proposition 2.5, and the two technical lemmas (Lemma 4.4 and Lemma 4.7) that carry the main conditional-moment bounds.","marker":"[10]"},{"why":"Provides the resolvent regularity result used to transform the SDE so that the drift error can be controlled by the difference of the transformed solutions.","marker":"[11]"},{"why":"Establishes the Gaussian (α=2) version of the randomised EM rate that Theorem 2.9 extends to α-stable noise.","marker":"[6]"},{"why":"States the stochastic sewing lemma that converts the conditional-moment bounds into pathwise strong-error estimates.","marker":"[19]"},{"why":"Introduces the randomised evaluation idea and the half-order convergence for irregular time coefficients that the scheme builds on.","marker":"[17]"},{"why":"Supplies the PDE-based EM error analysis that Section 4 extends from truncated to full symmetric α-stable processes.","marker":"[22]"}],"fun_headline_variants":["Randomised EM gains half order for stable SDEs with Hölder drift","Sampling drift randomly lifts strong order for heavy-tailed SDEs","Randomised EM beats standard EM for stable noise with Hölder drift","Strong order gain from random drift sampling in EM for α-stable SDEs","Randomised EM attains half-order gain over EM for heavy-tailed Lévy noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorem's rate depends on the paper's assertion that two technical lemmas proven for the standard Euler scheme carry over unchanged to the scheme with the extra random time point and its joint filtration; if that transfer fails, the claimed convergence rate does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Randomised EM gains half order for stable SDEs with Hölder drift","Sampling drift randomly lifts strong order for heavy-tailed SDEs","Randomised EM beats standard EM for stable noise with Hölder drift","Strong order gain from random drift sampling in EM for α-stable SDEs","Randomised EM attains half-order gain over EM for heavy-tailed Lévy noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001059,"raw_usage":{"total_tokens":4497,"prompt_tokens":1055,"completion_tokens":3442,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":671,"completion_tokens_details":{"reasoning_tokens":3343}},"tokens_in":671,"tokens_out":3442,"duration_ms":28862,"temperature":1.0,"reasoning_tokens":3343,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:09:51.745523+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Inspect the proof's hinge: verify directly that Lemma 4.4 and Lemma 4.7 of reference [10] apply to the randomised conditional process. Concretely, compute $\\|E_s[|\\phi^{(n)}_t - E_s\\phi^{(n)}_t|^2]\\|_{L^2(\\Omega)}$ from (20) and check whether the exponent $1+\\beta\\wedge(\\eta/\\alpha)-\\varepsilon$ and the stated $\\mathcal{F}_{(\\kappa_n(t)-1/n)\\vee 1}$-measurability of $\\phi^{(n)}_{\\kappa_n(t)}$ hold on the joint filtration; a single parameter triple $(\\alpha,\\beta,\\eta)$ satisfying (19) where either check fails refutes the proof of Theorem 2.9.","supporting_citations":[{"cited_title":"Stochastic flows for Lévy processes with Hölder drifts","cited_arxiv_id":null,"evidence_quote":"Provides the resolvent regularity result used to transform the SDE so that the drift error can be controlled by the difference of the transformed solutions."},{"cited_title":"Randomised Euler-Maruyama method for SDEs with H\\\"older continuous drift coefficient","cited_arxiv_id":"2501.15527","evidence_quote":"Establishes the Gaussian (α=2) version of the randomised EM rate that Theorem 2.9 extends to α-stable noise."},{"cited_title":"A stochastic sewing lemma and applications.Electron","cited_arxiv_id":null,"evidence_quote":"States the stochastic sewing lemma that converts the conditional-moment bounds into pathwise strong-error estimates."},{"cited_title":"Error analysis of randomized Runge-Kutta methods for differential equations with time-irregular coefficients.Comput","cited_arxiv_id":null,"evidence_quote":"Introduces the randomised evaluation idea and the half-order convergence for irregular time coefficients that the scheme builds on."},{"cited_title":"Strong rate of convergence for the Euler-Maruyama approx- imation of SDEs with Hölder continuous drift coefficient","cited_arxiv_id":null,"evidence_quote":"Supplies the PDE-based EM error analysis that Section 4 extends from truncated to full symmetric α-stable processes."}],"review_version":1}