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Randomised Euler-Maruyama Method for SDEs with H\"older Continuous Drift Coefficient Driven by $\alpha$-stable L\'evy Process

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2507.11429 v1 pith:KV5IXLDT submitted 2025-07-15 math.PR

Jianhai Bao , Haitao Wang , Yue Wu , Danqi Zhuang This is my paper · ORCID
classification math.PR MSC 65C3065C0560G5160H1060H3560L90
keywords randomisedEuler–Maruyamaα-stableLévyprocessHöldercontinuousdriftstrongorderofconvergencestochasticsewinglemmairregularheavy-tailednoisetime-inhomogeneousSDE
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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $$ E\Big[\sup_{0\le t\le 1}\big|X_t-$X^{{(n)}}$_t\big|^p\Big]\le C $n^{{-(1/2+\gamma-\varepsilon)p}}$ $$ for 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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

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.

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 (4)
  1. [Proposition 3.5] 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.
  2. [Proposition 3.7] 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.
  3. [Section 4] 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.
  4. [Theorem 2.9] 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.
minor comments (5)
  1. [Abstract and Introduction] The phrase 'symmetric α-table process' in the abstract should read 'symmetric α-stable process'.
  2. [Proposition 3.6] 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).
  3. [Lemma 3.2] 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.
  4. [Remark 2.10] 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.
  5. [Global] There are several typographical errors throughout, including 'Mckean-Valsov', 'equiped', and 'stbale'; a careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the rate is derived from SDE structure via external lemmas, with only a non-load-bearing self-citation of [6].

full rationale

No circularity in the central derivation. Theorem 2.9's rate gamma = beta ∧ (eta/alpha) ∧ (1/2) is not put in as an assumption; it emerges from Lemmas 3.3 and 3.4, where the exponents beta ∧ (eta/alpha) come from the Hölder regularity of b and the scaling of the alpha-stable noise, and from Propositions 3.5–3.7, which combine these with the stochastic sewing lemma (Theorem 2.6) and its conditional shifted version (Theorem 2.7). The heavy external inputs—[10, Lemma 4.4 and 4.7] and [11, Corollary 2.10]—are from other research groups and are used as lemma-level tools, not as restatements of the target convergence rate; no fitted parameter is renamed as a prediction. The only self-citation is [6], the authors' Gaussian-noise randomised EM paper, used in the introduction and Remark 2.10 for motivation and comparison; the alpha-stable proof does not invoke [6], so the self-citation is not load-bearing. We therefore see no circular step. For completeness, a separate correctness concern exists: in Proposition 3.6, the J2 (and J4) estimates replace g2(X_{s1}) - g2(X_{s2}) by 2∥g2∥∞, yielding a C n^{-1}|t-s| term; since the power of |t-s| is exactly 1, this does not visibly satisfy the shifted-SSL hypothesis (16), which requires powers strictly greater than 1. This is an omitted verification/possible gap, not a circularity, and does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The theorem is a statement about a numerical scheme; it depends on external results: the stochastic sewing lemmas, the stable-semigroup estimates of [10], the Zvonkin-type resolvent bound of [11], and, critically, two unverified transfers of [10]'s lemmas to the randomised setting. There are no fitted constants and no invented entities; all method parameters are inputs. The dependency on the authors' own companion paper [6] is motivational and does not enter the derivation, so circularity is low.

assumptions (4)
  • standard math The stochastic sewing lemma (Theorem 2.6) and its conditional shifted version (Theorem 2.7) hold for the joint probability space of the Lévy noise and the i.i.d. uniform randomisation times.
    Theorem 2.6 is quoted from Lê [19, Theorem 2.4] and Theorem 2.7 from Butkovsky et al. [10, Lemma 3.1]; they are the backbone of all estimates in Section 3.
  • domain assumption Assumptions 2.2-2.4 (gradient estimates of the stable semigroup, generator estimates, and moment asymptotics) hold for the driving symmetric α-stable process.
    These are imposed as standing assumptions in Section 2.2; for symmetric α-stable processes they are known and verified in [10], but the paper does not prove them.
  • standard math The Zvonkin-type resolvent equation (17) has a solution u satisfying the gradient bound (18) of Theorem 2.8, cited from Chen-Song-Zhang [11, Cor. 2.10].
    Invoked in Section 4 to construct M^λ_t and to ensure the contraction (29) for large λ.
  • ad hoc to paper The external estimates [10, Lemma 4.4] and [10, Lemma 4.7] are applicable to the randomised scheme's conditional processes with the paper's parameter choices (θ=η, τ=1+β∧(η/α)−ε, ε0=1, γ=1/2).
    The paper asserts this applicability in Proposition 3.5 ('ensure the applicability of [10, Lemma 4.4]') and Proposition 3.6 ('by [10, Lemma 4.7]'), but does not verify the lemmas' hypotheses in the randomised setting; the main rate rests on these bounds.

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Pith. "Pith review of Randomised Euler-Maruyama Method for SDEs with H\"older Continuous Drift Coefficient Driven by $\alpha$-stable L\'evy Process." pith.science (2026). https://pith.science/paper/KV5IXLDT

@misc{pith2026250711429,
  author       = {Pith},
  title        = {Pith review of: Randomised Euler-Maruyama Method for SDEs with H\"older Continuous Drift Coefficient Driven by $\alpha$-stable L\'evy Process},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KV5IXLDT}},
  note         = {Machine review of arXiv:2507.11429}
}
abstract

In this paper, we examine the performance of randomised Euler-Maruyama (EM) method for additive time-inhomogeneous SDEs with an irregular drift driven by symmetric $\alpha$-table process, $\alpha\in (1,2)$. In particular, the drift is assumed to be $\beta$-H\"older continuous in time and bounded $\eta$-H\"older continuous in space with $\beta,\eta\in (0,1]$. The strong order of convergence of the randomised EM in $L^p$-norm is shown to be $1/2+(\beta \wedge (\eta/\alpha)\wedge(1/2))-\varepsilon$ for an arbitrary $\varepsilon\in (0,1/2)$, higher than the one of standard EM, which cannot exceed $\beta$. The result for the case of $\alpha \in (1,2)$ extends the almost optimal order of convergence of randomised EM obtained in (arXiv:2501.15527) for SDEs driven by Gaussian noise ($\alpha=2$), and coincides with the performance of EM method in simulating time-homogenous SDEs driven by $\alpha$-stable process considered in (arXiv:2208.10052). Various experiments are presented to validate the theoretical performance.

Figures

Figures reproduced from arXiv: 2507.11429 by the authors.

Figure 1
Figure 1. Semi-log plots of error vs. stepsize for Example 1 (the Weierstrass function). [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Semi-log plots of error vs. stepsize for Example 2 (the jump-type drift). [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Semi-log plots of error vs. stepsize for Example 3 (the piecewise root function). [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Semi-log plots of error vs. stepsize for Example 4 (the temporal highly-oscillated function). [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]

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Reviewed August 6, 2026 · model on record in the stance chip above.