Randomised Euler-Maruyama method for SDEs with H\"older continuous drift coefficient

Jianhai Bao; Yue Wu

REVIEW 3 major objections 5 minor 1 cited by

Randomised Euler-Maruyama method for SDEs with H\"older continuous drift coefficient

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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stochastic-sewing argument; the main fragility is an imported PDE bound and an 'almost optimal' claim resting on unpublished work. read the letter →

arxiv 2501.15527 v1 pith:DVEA4K2T submitted 2025-01-26 math.PR cs.NAmath.NA

Jianhai Bao , Yue Wu This is my paper · ORCID

classification math.PRcs.NAmath.NA MSC 65C3065C0560H1060H3560L90

keywords randomisedEuler-MaruyamastochasticdifferentialequationssewinglemmaHöldercontinuousdriftstrongconvergenceorderadditivenoisequadraturemartingaleerrorsequence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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 $\varepsilon$.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $1/2+\gamma$.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $\varepsilon$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

My 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[§3, Lemma 3.7 and Eq. (3.17)] 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.
[§4, Lemma 4.1] 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.
[§4, Eqs. (4.5)–(4.8)] 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.

minor comments (5)

[§3, Eq. (3.4)] 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.
[§4, proof of Theorem 4.2] The phrase 'the contrain on ε' should read 'the constraint on ε'.
[§4, Theorem 4.2] 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.
[§3, Eq. (3.19)] 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|.
[Appendix B] In the proof of the alternative martingale bound, 'integrating with respect to P^W' should be 'integrating with respect to P^B'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the randomized EM rate is derived from external stochastic-sewing and PDE estimates, not from its own conclusion.

full rationale

The derivation chain is self-contained in the relevant sense. The central rate in Theorem 4.2 is obtained by combining new quadratic bounds (Propositions 3.2 and 3.4, Lemma 3.7, Corollary 3.8) with an imported backward-Kolmogorov estimate (Lemma 4.1, cited from the external paper [15]). None of these steps fits a parameter to the target error or defines the claimed rate into the assumptions. The randomized scheme (1.12) is motivated by earlier randomized-quadrature work, including papers co-authored by Wu, but the main proof does not cite those papers as the source of the convergence estimate; the quadratic bounds are proven here via the stochastic sewing lemma and heat-kernel estimates from [3] and [14]. The self-citations are contextual, not load-bearing. Lemma 4.1 is indeed quoted rather than proved in this paper, so its uniformity and validity are a proof-support or correctness concern, but an unproved external lemma is not circularity. Similarly, Remark 4.3's 'almost optimal' claim relies on an unpublished lower bound [7]; that is a verifiability gap, not a circular reduction. No equation or fitted quantity is renamed as a prediction, and no uniqueness or ansatz is imported from the authors' own prior work to force the result. Therefore the honest finding is no circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

No fitted constants appear in the proof; the only small parameter is the arbitrary epsilon. The central claim imports standard stochastic analysis results as axioms, with Lemma 4.1 being the most specialized and load-bearing.

assumptions (5)

standard math Stochastic sewing lemma (Theorem 2.2 of [14]) is used as a black box to bound randomized quadrature errors in Section 3.
Invoked in Propositions 3.2, 3.4 and Lemma 3.3; the proof assumes the theorem and its constants rather than deriving them.
standard math PDE regularity estimates for the backward Kolmogorov equation (Lemma 4.1 of [15]) provide solutions u with controlled gradient and Hessian.
This is the most specialized imported result; Theorem 4.2's Zvonkin transform and contraction argument depend on it.
standard math Girsanov theorem with Novikov condition is applied pathwise in the randomization variables in Lemma 3.7.
Used to transfer Brownian motion quadratic bounds to the numerical solution; the drift must be bounded for the exponential moments to be finite.
standard math Heat kernel estimates from Proposition A.1 and A.2 of [3] control conditional expectations of Holder functions of Brownian motion.
Used in Proposition 3.2 to estimate the I3 term; quoted without proof.
domain assumption The drift f is assumed to lie in C^{alpha,beta}_b, i.e. bounded and alpha-Holder in time and beta-Holder in space.
This is the class of SDEs under study; boundedness is used for the Novikov condition, well-posedness, and uniform Lp moment bounds.

invented entities (1)

i.i.d. U(0,1) randomization variables tau_j inserted into the time grid independent evidence
purpose: Randomize the drift evaluation point inside each step so that quadrature errors become a martingale and gain half an order in time regularity.
Introduced in (1.12) and (1.13); the variables are artificial random input, but their effect is quantified by Theorem 4.2, which is a falsifiable mathematical prediction. They are not new to the literature, as prior randomized schemes use the same device.

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Pith. "Pith review of Randomised Euler-Maruyama method for SDEs with H\"older continuous drift coefficient." pith.science (2026). https://pith.science/paper/DVEA4K2T

@misc{pith2026250115527,
  author       = {Pith},
  title        = {Pith review of: Randomised Euler-Maruyama method for SDEs with H\"older continuous drift coefficient},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DVEA4K2T}},
  note         = {Machine review of arXiv:2501.15527}
}

abstract

In this paper, we examine the performance of randomised Euler-Maruyama (EM) method for additive time-inhomogeneous SDEs with an irregular drift. In particular, the drift is assumed to be $\alpha$-H\"older continuous in time and bounded $\beta$-H\"older continuous in space with $\alpha,\beta\in (0,1]$. The strong order of convergence of the randomised EM in $L^p$-norm is shown to be $1/2+(\alpha \wedge (\beta/2))-\epsilon$ for an arbitrary $\epsilon\in (0,1/2)$, higher than the one of standard EM, which is $\alpha \wedge (1/2+\beta/2-\epsilon)$. The proofs highly rely on the stochastic sewing lemma, where we also provide an alternative proof when handling time irregularity for a comparison.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Randomised Euler-Maruyama Method for SDEs with H\"older Continuous Drift Coefficient Driven by $\alpha$-stable L\'evy Process
math.PR 2025-07 conditional novelty 6.0 of 10

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.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [15]

Pamen, O.M. and Taguchi, D.: Strong rate of convergence for the Euler-Maruyama approximation of SDEs with H¨ older continuous drift coeﬃcient,Stochastic Processes and their Applications, 127(8), 2542-2559, (2017)

work page 2017

[2] [7]

and Yaroslavtseva, L.: Sharp lower error bounds for pathwise approximation of SDEs with a H¨ older drift coeﬃcient,In work progress , (2025)

Ellinger, S., M¨ uller-Gronbach, T. and Yaroslavtseva, L.: Sharp lower error bounds for pathwise approximation of SDEs with a H¨ older drift coeﬃcient,In work progress , (2025)

work page 2025

[3] [1]

Biswas, S., Kumar, C., Neelima, Reis, G.D. and Reisinger , C.: An explicit Milstein-type scheme for interacting particle systems and McKean-Vlasov SDEs with c ommon noise and non-diﬀerentiable drift coeﬃcients, The Annals of Applied Probability , 34(2), pp.2326-2363, (2024)

work page 2024

[4] [2]

Burkholder, D.L.: Martingale transforms, The Annals of Mathematical Statistics , 37(6), pp.1494- 1504, (1966)

work page 1966

[5] [3]

and Gerencs´ er, M.: Approx imation of SDEs: a stochastic sewing approach, Probability theory and related ﬁelds , 181, 975-1034, (2021)

Butkovsky, O., Dareiotis, K. and Gerencs´ er, M.: Approx imation of SDEs: a stochastic sewing approach, Probability theory and related ﬁelds , 181, 975-1034, (2021)

work page 2021

[6] [4]

Davie, A.M.: Uniqueness of solutions of stochastic diﬀerential equations, International Mathematics Research Notices, p.rnm124 (2007)

work page 2007

[7] [5]

and Lˆ e, K.: Quantifying aconvergence theorem of Gy¨ ongy and Krylov, The Annals of Applied Probability , 33(3), pp.2291-2323, (2023)

Dareiotis, K., Gerencs´ er, M. and Lˆ e, K.: Quantifying aconvergence theorem of Gy¨ ongy and Krylov, The Annals of Applied Probability , 33(3), pp.2291-2323, (2023)

work page 2023

[8] [6]

RANDOMISED EM FOR SDES WITH H ¨OLDER CONDITIONS 21

Daun, T.: On the randomized solution of initial value pro blems, Journal of Complexity , 27(3-4), pp.300-311, (2011). RANDOMISED EM FOR SDES WITH H ¨OLDER CONDITIONS 21

work page 2011