{"id":"c0795d84-fa56-4b8b-be91-d8fed8fd0d58","arxiv_id":"2501.15527","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Randomized Euler-Maruyama achieves strong Lp order 1/2 + min(alpha, beta/2) - epsilon for additive SDEs with alpha-Holder time and beta-Holder space drift, improving on standard Euler-Maruyama.","lead":"This paper proves that a randomized Euler-Maruyama scheme, which evaluates the drift at a random point inside each time step, converges at a strong rate of 1/2 + min(alpha, beta/2) minus epsilon for SDEs with Holder continuous drift. The rate beats standard Euler-Maruyama and is close to the best possible, making randomized time grids a useful tool for rough drift coefficients.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The recursion in Theorem 4.2 hinges on an imported gradient contractivity bound (Lemma 4.1) that the paper does not prove; if the sup-norm version fails or is not uniform, the partition argument collapses.","rationale":"After reviewing the quadratic estimates, the Girsanov transfer, and the PDE recursion, I agree with the reader that Lemma 4.1 is the load-bearing assumption. The stochastic sewing lemmas (Propositions 3.2-3.4) are internally consistent: the randomized time cancellation in (3.4) and the application of Theorem 2.2 check out. Lemma 3.7 has a constant error: the factor involving dP^B/dP^tildeB should be exponential in ||f||^2, not bounded by ||f||, and the lemma as stated for general g would be false at f=0. This is fixable and does not affect Theorem 4.2 because the lemma is only applied with g equal to f or its components. The main theorem's validity therefore rests on the imported gradient estimate. The proposed test isolates that estimate in the simplest nontrivial case. If the test passes, I would accept Theorem 4.2 as conditionally sound; if it fails, the rate proof has a genuine gap.","tokens_in":20540,"tokens_out":38008,"duration_ms":352421,"concrete_test":"Take f=0, phi(x)=eta(x)|x|^beta with a compactly supported smooth cutoff eta, and set u(t)=int_t^delta P_{r-t} phi dr on [0,delta], u(delta)=0. Compute ||grad u||_inf and the C^beta seminorm of grad u as delta goes to 0 for several beta in (0,1). Check whether both are bounded by C0 delta^{1/2} ||phi||_{C^beta} with C0 uniform in beta. If the sup-norm version fails, Theorem 4.2's contraction coefficient cannot be made small; if only the C^beta version fails, the proof can likely be repaired using the C^1 bound on D^2 u.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.2 is built on Lemma 4.1, quoted from [15]. The proof needs, on each subinterval, the bound ||grad u_{l,i}||_inf <= epsilon so that the factor (1+epsilon) and the epsilon-term in (4.4) are produced and the recursion can be closed. Lemma 4.1 asserts this via ||grad u||_{C^beta} <= C0 (T_{i+1}-T_i)^{1/2} ||phi||_{C^beta}, with the partition chosen so that C0 delta^{1/2} ||phi|| <= epsilon. This estimate is not derived here. If the constant C0 is not uniform as beta approaches 0, or if the bound only holds for phi smoother than C^beta, then the required partition may be impossible and Theorem 4.2 lacks a proof. I do not see an internal contradiction, but this is the step on which the central rate claim rests.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a randomised Euler-Maruyama scheme for additive SDEs with drift f in C^{α,β}_b, using i.i.d. U(0,1) shifts τ_j inside each time step. The main theorem (Theorem 4.2) asserts strong L^p convergence of order 1/2 + (α∧β/2) − ε, improving on the standard EM rate α∧(1/2+β/2−ε). The proof is built from two quadratic bounds established via the stochastic sewing lemma (Propositions 3.2 and 3.4), a Girsanov transfer from the numerical solution to shifted Brownian motion (Lemma 3.7), and a PDE-based error decomposition on a partition imported from [15] (Lemma 4.1). An alternative martingale proof of Lemma 3.3 is given in Appendix B.","tokens_in":20666,"tokens_out":24030,"duration_ms":204751,"significance":"The claimed rate is a genuine improvement for time-irregular drifts and is consistent with known lower bounds from randomized ODE quadrature (1/2+α) and SDE approximation with β-Hölder drift (1/2+β/2). The paper is the first to apply the stochastic sewing lemma to randomised schemes, and the alternative discrete-martingale proof in Appendix B is a useful comparison. However, Lemma 3.7 is false as stated because the Girsanov density factor is mishandled, and Lemma 4.1 is imprecisely stated and imported without proof; since both are used in the main recursion, the main theorem is not currently established. The errors appear fixable within the paper's scope, and the overall strategy is sound.","major_comments":[{"comment":"The stated bound (3.14) cannot hold with the factor ‖f‖_{C^{α,β}_b}. Taking f≡0 and g(t,x)=t gives a nonzero left-hand side (the randomised quadrature error ∫(r−κ^τ_n(r))dr has positive L^p norm under P^τ) while the right-hand side is zero. The error occurs in (3.17), where (E^B[dP_B/d\\tilde{P}_B|\\mathcal{F}^τ])^{1/2} is replaced by ‖f‖; in fact E^B[dP_B/d\\tilde{P}_B|\\mathcal{F}^τ]=1 because the density is a PB-martingale and \\mathcal{F}^τ is independent of B, so the correct second factor is 1. Please correct the statement and proof, and propagate the corrected constant through Corollary 3.8 and Theorem 4.2.","section":"§3, Lemma 3.7 and Eq. (3.17)"},{"comment":"This lemma is quoted from [15] without proof, and its first displayed condition cannot be a universal statement: for a fixed partition, ‖φ‖C0Δ^{1/2}≤ε cannot hold for all φ, since φ may have arbitrarily large C^β norm. The proof only needs the condition for φ=f_l, so the lemma should be restated with the partition chosen for f (or for the fixed φ), and the notation ‖φ‖_{C^β_b([0,1])} should be defined. Additionally, the line in the proof of Theorem 4.2 that bounds ‖∂u_{l,i}/∂x_k‖_{C^1_b} by Mε does not follow from the stated estimates: the C^1 norm is controlled by M‖φ‖_{C^β}, not by ε. Since these estimates generate the smallness of the ε-terms in the recursion, the authors should either prove Lemma 4.1 or give a precise citation and a corrected statement.","section":"§4, Lemma 4.1"},{"comment":"The passage from the BDG term ∫E|∇u_{l,i}(s,X_s)−∇u_{l,i}(s,X_s^{(n)})|^p ds to the Grönwall term ∫E sup_{T_{i-1}≤u≤s}|X_u−X_u^{(n)}|^p ds requires a Lipschitz bound on ∇u_{l,i}. Lemma 4.1 only records a C^β bound on ∇u; the needed Lipschitz property follows from u∈C^{2,β'} but is not stated or proved. As written, the displayed recursion does not close. Please add the explicit bound |∇u(x)−∇u(y)|≤M‖φ‖_{C^β}|x−y| and adjust the constants accordingly.","section":"§4, Eqs. (4.5)–(4.8)"}],"minor_comments":[{"comment":"In Eq. (3.4), the symbol f appears where g is meant in the second and third displayed lines; this is confusing and should be corrected.","section":"§3, Eq. (3.4)"},{"comment":"The phrase 'the contrain on ε' should read 'the constraint on ε'.","section":"§4, proof of Theorem 4.2"},{"comment":"Theorem 4.2 states p≥1 but the proof treats p≥2; add a sentence explaining that the case 1≤p<2 follows from the L^2 result by Lyapunov's inequality.","section":"§4, Theorem 4.2"},{"comment":"Eq. (3.19) is missing the integral sign and the differential dr inside the supremum; it should read sup_{0≤s≤t}|∫_0^s (g1(r,X_r)−g1(κ^τ_n(r),X_{κ_n(r)})) g2(r,X_r) dr|.","section":"§3, Eq. (3.19)"},{"comment":"In the proof of the alternative martingale bound, 'integrating with respect to P^W' should be 'integrating with respect to P^B'.","section":"Appendix B"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the journal's scope and the core idea is attractive. The errors in Lemma 3.7 and Lemma 4.1 are load-bearing but appear reparable without changing the main strategy; I would not reject on these grounds. The authors should carefully check the citation to [15]: they rely on a stronger PDE estimate than the standard one, so they should either prove it or quote the exact statement. The reference list contains a corrupted entry for [16] ('Przyby/suppress lowicz') that must be fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid, useful paper. The main theorem—strong order 1/2 + γ − ε for randomized EM on additive SDEs with (α, β)-Hölder drift—is new and improves on standard EM's α ∧ (1/2 + β/2 − ε) in the regime where time irregularity hurts. The proof strategy is well chosen: stochastic sewing for the quadratic error bounds, Girsanov to transfer from a Brownian driver to the numerical scheme, and a PDE transformation to close the recursion. Appendix B's discrete-martingale proof of Lemma 3.3 is a nice alternative that makes the mechanism transparent. The paper is clearly written and the technical estimates in Section 3 look right to me.\n\nThe soft spots are real but mostly peripheral. First, the proof of Theorem 4.2 rests on Lemma 4.1, quoted from Pamen–Taguchi, and that lemma is doing a lot of work: the partition of [0,1] has to make the gradient of the PDE solution small in C^β, uniformly in β and in the data. The paper doesn't prove it, and if that estimate fails in the stated form the main rate collapses. I don't see evidence it fails—[15] is a published source—but a referee should check the uniformity claim carefully. Second, the Girsanov density bound in Lemma 3.7 is stated too quickly; as written, the bound on E[dPB/dP̃B] looks too small. It's fixable by absorbing constants, and it doesn't affect the rate, but it needs correcting. Third, Remark 4.3's 'almost optimal' uses an unpublished lower bound [7] plus a heuristic splice of two lower bounds; that's a reasonable conjecture but not part of the theorem. Finally, there are typos (a mangled author name in the references, a missing integral in Corollary 3.8) that suggest the manuscript needs a proofreading pass.\n\nMy take: the central claim is likely correct and the paper deserves a serious referee. I'd send it to review with a request to verify Lemma 4.1's applicability, fix the Girsanov display, and tone down the 'almost optimal' remark.","headline":"A genuinely new strong-order rate for randomized EM on additive SDEs with Hölder drift, built on a coherent stochastic-sewing argument; the main fragility is an imported PDE bound and an 'almost optimal' claim resting on unpublished work.","tokens_in":21266,"tokens_out":4505,"would_cite":true,"duration_ms":38845,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65C30","65C05","60H10","60H35","60L90"],"pacs":[],"model":"deepseek-v4-flash","headline":"Randomized Euler-Maruyama for additive SDEs with $\\alpha$-Hölder time drift and bounded $\\beta$-Hölder spatial drift achieves strong $L^p$ order $1/2+(\\alpha\\wedge(\\beta/2))-\\varepsilon$, beating standard Euler when time irregularity is…","keywords":["randomised Euler-Maruyama","stochastic differential equations","stochastic sewing lemma","Hölder continuous drift","strong convergence order","additive noise","randomised quadrature","martingale error sequence"],"falsifier":"For the scalar SDE $dX(t)=|t-1/2|^\\alpha\\,dt+dB(t)$ with $\\alpha=0.4$, $\\beta=1$, the exact solution is $X(t)=x_0+\\int_0^t |s-1/2|^\\alpha ds+B(t)$, and the theorem predicts randomized Euler-Maruyama strong error of order $0.9-\\varepsilon$ while standard EM gives $0.4$. Simulating both on the same Brownian path and regressing $\\log E[\\sup_t|X_t-X^{(n)}_t|]$ against $\\log n$ for $n=2^9$ to $2^{15}$ should show those two slopes; a randomized slope at or below $0.4$ would falsify Theorem 4.2.","tokens_in":20263,"feed_emoji":"🎲","tokens_out":18431,"duration_ms":144741,"temperature":0.7,"pith_summary":"This paper studies a randomized Euler-Maruyama scheme for additive stochastic differential equations $dX(t)=f(t,X(t))dt+B(t)$ whose drift is $\\alpha$-Hölder in time and bounded $\\beta$-Hölder in space. The scheme evaluates the drift at a random point inside each grid cell, using i.i.d. uniform random variables independent of the Brownian motion. The paper proves that this randomisation lifts the strong $L^p$ convergence order to $1/2+(\\alpha\\wedge(\\beta/2))-\\varepsilon$ for any $p\\ge1$ and $\\varepsilon\\in(0,1/2)$, which is never worse and strictly better than the standard Euler-Maruyama order $\\alpha\\wedge(1/2+\\beta/2-\\varepsilon)$ whenever time irregularity is the bottleneck. The argument rests on the stochastic sewing lemma, a change of measure, and a PDE estimate for the backward Kolmogorov equation. The paper thereby claims that a one-line modification of the Euler scheme recovers a half order of convergence for rough time-dependent drifts, reaching the known optimal order up to an arbitrarily small $\\varepsilon$.","feed_headline":"Random time shifts give Euler-Maruyama a half-order speedup","feed_subtitle":"For α-Hölder time / β-Hölder space drift, the L^p error decays like n^{-(1/2+α∧β/2−ε)}, beating standard EM.","key_machinery":"The load-bearing object is the randomized Euler-Maruyama scheme itself, $X^{(n)}_t=x_0+\\int_0^t f(\\kappa^\\tau_n(s),X^{(n)}_{\\kappa_n(s)})\\,ds+B_t$, where $\\kappa^\\tau_n(s)=(\\lfloor ns\\rfloor+\\tau_{\\lfloor ns\\rfloor})/n$ and $(\\tau_i)$ are i.i.d. $U(0,1)$ random variables independent of the Brownian motion $B$. The mechanism that carries the argument is the martingale structure of the quadrature error sequence: on each grid cell, $\\int_{j/n}^{(j+1)/n} [g(r,B_{\\kappa_n(r)})-g(\\kappa^\\tau_n(r),B_{\\kappa_n(r)})]\\,dr$ is an $L^p$-martingale in the random-shift filtration, whose maximum is controlled by its quadratic variation. That is why the time-irregularity exponent $\\alpha$ appears with an added $1/2$. Around this, the proof uses the stochastic sewing lemma (Theorem 2.2) to promote local quadratic bounds of order $|t-s|^{1/2+\\varepsilon}n^{-(1/2+\\gamma-\\varepsilon)}$ to global supremum bounds, and a backward Kolmogorov equation whose solution's gradient is contractive in the interval length, which lets the error recursion be closed on a partition.","core_discovery":"The paper's central claim is Theorem 4.2: for $f\\in C^{\\alpha,\\beta}_b$ and the randomized Euler-Maruyama scheme (1.12), the strong error satisfies $E[\\sup_{0\\le t\\le1}|X(t)-X^{(n)}_t|^p]\\le C n^{-(1/2+\\gamma-\\varepsilon)p}$ for every $p\\ge1$ and $\\varepsilon\\in(0,1/2)$, with $\\gamma:=\\alpha\\wedge(\\beta/2)$. In words, the $p$-th-mean supremum error decays like $n^{-(1/2+\\gamma-\\varepsilon)}$. The randomized evaluation at $t_{j-1}+\\tau_j/n$ converts the deterministic time-quadrature error into a discrete martingale in the artificial randomness, which buys an extra $n^{-1/2}$. The error analysis splits the scheme error into two quadratic bounds, proves them through the stochastic sewing lemma after an equivalent change of measure, and closes the recursion using the backward Kolmogorov equation on a partition where the gradient of its solution is uniformly small. The authors also argue the rate is almost optimal, since the ODE lower bound $1/2+\\alpha$ and the SDE lower bound $1/2+\\beta/2$ combine to the target $1/2+\\gamma$.","pith_inferences":["Not claimed in the paper, but the same random-shift mechanism should transfer to Milstein-type schemes with non-differentiable drift, where time irregularity also enters as a quadrature term and the half-order gain should persist.","A cheap stress test: for a purely time-dependent drift $f(t)$, the SDE reduces to a deterministic ODE plus additive Brownian motion, so the theorem predicts that randomized EM matches the randomized-ODE quadrature rate $1/2+\\alpha$; a scalar benchmark with $f(t)=|t-1/2|^\\alpha$ isolates that mechanism.","Whether the $\\varepsilon$-loss is removable is left open; if a matching lower bound for the randomized scheme could be proved, the rate $1/2+\\gamma$ would be sharp for this numerical method rather than merely almost optimal."],"forward_implications":["When $\\alpha<1/2$, the theorem gives a strong rate of $1/2+\\alpha-\\varepsilon$, half an order above the standard Euler rate $\\alpha$, so the gain is largest exactly when time irregularity is the bottleneck.","When $\\beta<2\\alpha$, the rate becomes $1/2+\\beta/2-\\varepsilon$, independent of the time-Hölder exponent; spatial roughness then controls the error.","The bound holds for every $p\\ge1$ and controls the supremum of the error over the whole interval, not just the error at a fixed time.","The theorem is almost optimal: the lower bounds $1/2+\\alpha$ for randomized ODE quadrature and $1/2+\\beta/2$ for additive SDEs with $\\beta$-Hölder drift put the target at $1/2+\\gamma$, which the randomized EM attains up to an arbitrarily small $\\varepsilon$."],"supporting_citations":[{"why":"Supplies the PDE strategy and the backward-Kolmogorov contractivity estimate (Lemma 4.1) used to close the main error recursion.","marker":"[15]"},{"why":"Provides the heat-kernel estimates and the Brownian-motion quadratic bounds that the stochastic sewing arguments start from.","marker":"[3]"},{"why":"States the stochastic sewing lemma, the mechanism that converts local quadratic bounds into global supremum error estimates.","marker":"[14]"},{"why":"Introduces the randomized quadrature rule and the discrete-martingale error analysis for ODEs with time-irregular coefficients on which the scheme is modelled.","marker":"[12]"},{"why":"Gives the ODE lower bound $1/2+\\alpha$ for randomized algorithms, one of the two ingredients in the near-optimality remark.","marker":"[11]"},{"why":"Provides a sharp lower error bound for strong approximation with Hölder drift, supporting the almost-optimality claim.","marker":"[7]"}],"fun_headline_variants":["Random time shifts give Euler-Maruyama a half-order boost","Randomized EM outruns standard for Hölder drift","Irregular drift? Randomize EM's time grid for extra order","Half-order faster: randomized time stepping for rough SDEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument collapses if the quoted backward-Kolmogorov estimate of Lemma 4.1—that the solution's spatial gradient, measured in the $\\beta$-Hölder norm, is bounded by a constant times the square root of the time-interval length—fails or is not uniform in the stated way.","fun_headline_variants_meta":{"raw":{"variants":["Random time shifts give Euler-Maruyama a half-order boost","Randomized EM outruns standard for Hölder drift","Irregular drift? Randomize EM's time grid for extra order","Half-order faster: randomized time stepping for rough SDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000599,"raw_usage":{"total_tokens":2826,"prompt_tokens":994,"completion_tokens":1832,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":610,"completion_tokens_details":{"reasoning_tokens":1761}},"tokens_in":610,"tokens_out":1832,"duration_ms":12730,"temperature":1.0,"reasoning_tokens":1761,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:13:37.727074+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the scalar SDE $dX(t)=|t-1/2|^\\alpha\\,dt+dB(t)$ with $\\alpha=0.4$, $\\beta=1$, the exact solution is $X(t)=x_0+\\int_0^t |s-1/2|^\\alpha ds+B(t)$, and the theorem predicts randomized Euler-Maruyama strong error of order $0.9-\\varepsilon$ while standard EM gives $0.4$. Simulating both on the same Brownian path and regressing $\\log E[\\sup_t|X_t-X^{(n)}_t|]$ against $\\log n$ for $n=2^9$ to $2^{15}$ should show those two slopes; a randomized slope at or below $0.4$ would falsify Theorem 4.2.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the PDE strategy and the backward-Kolmogorov contractivity estimate (Lemma 4.1) used to close the main error recursion."},{"cited_title":"and Gerencs´ er, M.: Approx imation of SDEs: a stochastic sewing approach, Probability theory and related fields , 181, 975-1034, (2021)","cited_arxiv_id":null,"evidence_quote":"Provides the heat-kernel estimates and the Brownian-motion quadratic bounds that the stochastic sewing arguments start from."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the stochastic sewing lemma, the mechanism that converts local quadratic bounds into global supremum error estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the randomized quadrature rule and the discrete-martingale error analysis for ODEs with time-irregular coefficients on which the scheme is modelled."},{"cited_title":"and Milla, B.: The randomized complexity o f initial value problems, Journal of Com- plexity, 24(2), pp.77-88 (2008)","cited_arxiv_id":null,"evidence_quote":"Gives the ODE lower bound $1/2+\\alpha$ for randomized algorithms, one of the two ingredients in the near-optimality remark."},{"cited_title":"and Yaroslavtseva, L.: Sharp lower error bounds for pathwise approximation of SDEs with a H¨ older drift coefficient,In work progress , (2025)","cited_arxiv_id":null,"evidence_quote":"Provides a sharp lower error bound for strong approximation with Hölder drift, supporting the almost-optimality claim."}],"review_version":1}