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The adaptive EM schemes for McKean-Vlasov SDEs with common noise in finite and infinite horizons

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Pith's one-line read An adaptive Euler–Maruyama scheme for McKean–Vlasov SDEs with common noise attains strong L^p convergence order 1/2 in the step-size parameter, on both finite and infinite horizons, even when drift and diffusion grow superlinearly.

desk verdict Useful extension of adaptive EM to common-noise McKean-Vlasov, but the main theorems overclaim the p-range and the numerical section has no numbers. read the letter →

arxiv 2509.00521 v1 pith:WMILPNXM submitted 2025-08-30 math.NA cs.NAmath.PR

Hu Liu , Shuaibin Gao , Junhao Hu This is my paper · ORCID
classification math.NAcs.NAmath.PR MSC 60H3565C30
keywords McKean–VlasovSDEscommonnoiseadaptiveEuler–MaruyamasuperlineargrowthstrongconvergenceinfinitehorizonparticlesystemWassersteindistance
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 that a simple adaptive Euler–Maruyama scheme can reliably simulate McKean–Vlasov SDEs with common noise when the drift and diffusion coefficients grow faster than linearly. The central result is an L^p strong error bound of order 1/2 in the adaptive step-size parameter δ, uniformly over all particles, for both finite time intervals and, under a stronger dissipativity condition, uniformly for all times. Fixed-step Euler schemes generally diverge for superlinear coefficients, so the adaptive step is the device that keeps moments bounded. A reader should care because these equations model large interacting systems—financial networks, neurons—where all particles share a common noise, and the paper supplies explicit convergence rates for this setting.

What carries the argument

The central object is the adaptive timestep function hδ(x, µ) = δ min(T, h(x, µ)) (or δ min(hmax, h(x, µ)) on the infinite horizon), where h(x, µ) = C_h / (1 + |b(x,µ)| |σ(x,µ)| + |b(x,µ)| |σ0(x,µ)| + |x|^q)^2, together with the lower bound (3.5) that prevents steps from collapsing to zero. At each step the scheme advances by the minimum step h^min_n across all N particles, evaluated at the current particle values and empirical measure. The upper form of h bounds the growth of the EM increments, while the lower bound guarantees finite steps and controls the local error; Itô's formula, Young's and Hölder's inequalities, and Gronwall's lemma then convert these bounds into the δ^{p/2} strong er

What would settle it

Take the one-dimensional example with b(x,µ) = −2x − 3x^2|x| − 2∫xµ(dx), σ(x,µ) = σ0(x,µ) = ¼(1 + |x|^{3/2} + ∫xµ(dx)), and hδ = δ min(1, (3|x|^3 + 2∫|x|^2µ(dx))^{-1}). With N = 10^4 particles, compute max_i E|Xhat^{i,N}_T − X^{i,N}_T|^2 for δ = 2^{-l} at T = 1. If the log-log slope of this error versus δ is not close to 1 (the p=2 case of δ^{p/2})—or if, in the infinite-horizon setting, the error grows with time for fixed δ despite λ1 > λ2 + 5/2—the central rate claim fails.

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Extended reading notes

Core claim

For an interacting particle system associated with a McKean–Vlasov SDE with common noise, under a monotonicity/polynomial-growth condition (Assumption 3.1) and an adaptive step function satisfying Assumption 3.2, the adaptive EM scheme (3.1) satisfies max_{i≤N} sup_{0≤t≤T} E|Xhat^{i,N}_t − X^{i,N}_t|^p ≤ C δ^{p/2} (Theorem 3.2). Under stronger dissipativity (Assumption 3.3) and bounded adaptive steps (Assumption 3.4), the same bound holds uniformly in t ≥ 0 (Theorem 3.4). The constants depend on T, p, and the coefficients but not on N; the number of steps per particle is O(δ^{-1}). The paper also records propagation-of-chaos estimates in L^p and on the infinite horizon, so the total mean-fie

Load-bearing premise

The load-bearing premise is that one can actually write down a positive adaptive step function h(x, µ) satisfying both the lower bound (3.5) and the reciprocal-polynomial upper form in Assumptions 3.2 or 3.4; the convergence theorems only apply to such explicitly constructed step functions, and constructing them for general superlinear coefficients is not automatic.

Editorial extensions

If this is right

  • The strong error bound max_i sup_{0≤t≤T} E|Xhat^{i,N}_t − X^{i,N}_t|^p ≤ C δ^{p/2} makes the adaptive EM scheme a viable replacement for tamed or projected schemes when coefficients are superlinear and common noise is present.
  • Because the expected number of steps per particle is O(δ^{-1}), the computational cost scales like δ^{-1} per path, comparable to fixed-step EM, while preserving convergence.
  • On the infinite horizon, the bound is uniform in time, so long-time simulation of mean-field models with common noise can be run without the error growing with t, provided λ1 > λ2 + 5/2.
  • Combined with the propagation-of-chaos estimates (Propositions 3.2 and 3.3), the total particle-plus-time error splits into an N-dependent term and a δ^{p/2} term, giving a concrete recipe for balancing N and δ in simulations.

Reading between the lines

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

  • The paper gives no numerical tables; an immediate test is to reproduce Examples 4.1 and 4.2 and estimate the empirical order of the RMSE versus δ—if the order is not close to 1/2, the constants or assumptions would need scrutiny.
  • Using the minimum step across all particles is conservative: a per-particle asynchronous adaptive step would reduce idle waiting, and the error analysis suggests the common-noise Brownian increment is what forces the min; testing asynchronous variants is a natural extension.
  • The sufficient condition λ1 > λ2 + 5/2 for the infinite-horizon rate is stronger than the natural stability condition λ1 > λ2; the extra 5/2 may be an artifact of the proof, and sharper local-error estimates could lower it.
  • For multilevel Monte Carlo, an L^p strong rate of 1/2 means the variance of level differences should decay like δ; this paper does not discuss MLMC, but the uniform particle-wise error control is exactly what MLMC estimators need.
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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 studies an adaptive Euler-Maruyama scheme for McKean-Vlasov SDEs with common noise, where the drift and diffusion coefficients can grow superlinearly in the state variable. The scheme (3.1) uses a common minimum step h_min over all particles. Under monotonicity and polynomial-growth assumptions (Assumptions 3.1–3.4), the authors prove finite-horizon moment bounds (Theorem 3.1), finite-horizon strong L^p convergence of the numerical particle system to the interacting particle system with rate δ^{p/2} (Theorem 3.2), an expected step-count bound (Proposition 3.1), and analogous infinite-horizon moment and convergence results (Theorems 3.3 and 3.4). Propagation-of-chaos estimates from the literature are invoked to convert the particle-system error into an error against the limiting McKean-Vlasov SDE. Two numerical examples are described in Section 4, but no numerical output is reported.

Significance. If the stated results hold, the paper would give a useful extension of adaptive EM schemes to McKean-Vlasov SDEs with common noise, covering superlinear drift and diffusion in both finite and infinite horizons. The claimed δ^{p/2} strong L^p rate is the expected order for an adaptive EM scheme, and the paper provides explicit step-size constructions and separates the finite- and infinite-horizon analyses. Strengths include the explicit form of the step-size function, the moment estimates for the numerical solution, and the use of published propagation-of-chaos results to combine step-size and particle-number errors. However, the central convergence theorems are stated for a wider range of p than the proofs support, and a key assumption on the step-size function is mis-stated. These issues are load-bearing and require correction before the claims can be accepted.

major comments (4)
  1. [Theorem 3.2 and its proof] The theorem states convergence for any 2 ≤ p < \tilde p, but the proof requires moments of the numerical solution of order strictly larger than \tilde p. After Eq. (3.11), the local-error estimate for E|\hat X_s - \bar X_s|^{2p} uses E|b(\bar X)|^{4p}, E||σ(\bar X)||^{4p}, and E||σ_0(\bar X)||^{4p}. By Remark 3.1 these are controlled only if 4p(q+1) < \tilde p and 2p(q+2) < \tilde p. The later bounds on E[\tilde Q_L^{2p}] and E[\tilde Q_K^{2p}] additionally require 2pq < \tilde p. Theorem 3.1 only guarantees moments of order strictly below \tilde p, so the proof does not close for all p < \tilde p. The statement must be restricted to p satisfying these higher-moment conditions, or the moment estimates must be strengthened.
  2. [Theorem 3.4 and its proof] The statement allows p > 0, but the proof is written entirely for p ≥ 2. The Itô formula applied to |e_t|^p uses the factor |e_s|^{p-2}, the martingale terms use |e_s|^{p-2} ⟨e_s,·⟩, and Young's inequality is applied with exponents that require p ≥ 2. Assumption 3.3 is also formulated with the coefficient p−1, and the proof uses p(p−1). For 0 < p < 2 these steps are not justified as written, and the singularity of |e|^{p−2} at zero is not addressed. Restrict the theorem to 2 ≤ p < \tilde p (with the additional moment restrictions identified above) or supply a separate argument for p < 2.
  3. [Assumptions 3.2(1) and 3.4(1)] The displayed formula h(x,μ) = C_h (1 + |b||σ| + |b||σ0| + |x|^q)^2 is inconsistent with the rest of the paper. Remark 3.2 and the proof of Theorem 3.1 rely on bounds such as |b|^2 h ≤ C and |b||σ| h^{1/2} ≤ C, which require h to be of order 1/(1 + |b||σ| + |b||σ0| + |x|^q)^2. The examples in Section 4 also use h = (1 + ⋯)^{-2}. As written, Assumptions 3.2(1) and 3.4(1) give the reciprocal and contradict Eq. (3.6). Replace these formulas by h = C_h / (1 + |b||σ| + |b||σ0| + |x|^q)^2 in both assumptions.
  4. [Section 4] The section is titled 'Numerical examples' and the abstract advertises an illustration of the theoretical results, but no numerical results are reported. There are no tables, figures, RMSE values, convergence-rate estimates, or CPU/time-step statistics. The section only defines two model problems and admissible adaptive step functions. Either add the actual numerical experiments and compare the observed rates with the predicted δ^{1/2} order, or revise the abstract and introduction to state that only admissible step-size constructions are given.
minor comments (5)
  1. [Proof of Theorem 3.2, Eq. (3.10)] The quantity \tilde Q_K is defined as K(1+|x|^q+|y|^q), but in line (3.10) it is squared. Align the notation, e.g. define \tilde Q_K := K(1+|x|^q+|y|^q)^2 or remove the square in (3.10).
  2. [Theorem 3.4, first display] In the SDE for e_t, the coefficient of dW^0_t should be σ0(\bar X, μ) − σ0(X, μ), not σ(...) − σ(...). The same typo appears at the beginning of the proof.
  3. [Assumptions 3.2 and 3.4] The step-size function is written as h : R^d → R_+, but it depends on the measure argument μ as well. It should be h : R^d × P_2(R^d) → R_+.
  4. [Abstract] The abstract says the convergence rates 'react to the particle number and step size'. Theorems 3.2 and 3.4 give only the δ^{p/2} step-size rate; the particle-number rate appears only in the separate propagation-of-chaos propositions. Clarify this distinction.
  5. [Theorem 3.3 proof, Eq. (3.13)] The line E[W_2^p(μ,δ_0)] = E|\bar X|^p is not generally true; if the Wasserstein exponent intended is p, use W_p, or explain that only a second-moment bound is being used.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central convergence proofs are self-contained, and the self-citations appear only in auxiliary, non-load-bearing positions.

full rationale

The main convergence claims, Theorems 3.2 and 3.4, are proved directly from the stated Assumptions 3.1-3.4 without invoking the paper's own prior results. In each proof, the error process e_i = hatX^{i,N} - X^{i,N} is written as an Itô process, controlled by the monotonicity/polynomial-growth conditions, and the local error term E|hatX_s - barX_s|^{2p} is bounded by C delta^p using the explicit adaptive step-size bounds in Assumptions 3.2/3.4. The moment bounds used in these proofs come from Theorems 3.1 and 3.3, which are proved earlier in the same paper from the same assumptions. Thus the order-1/2 convergence in delta is derived, not assumed or fitted. No parameter is calibrated to the target error, and the two numerical examples are illustrations rather than evidence used to justify the theorems. The self-citations ([13], [26]) appear in Propositions 3.2 and 3.3, which state propagation-of-chaos estimates between the noninteracting system and the particle system. Those propositions are not used in the proofs of Theorems 3.2 or 3.4, so they are not load-bearing for the central derivation. The skeptical concern about the p-range (the local-moment estimates require moments of order 4p(q+1) that may exceed the range guaranteed by Theorem 3.1) is a potential correctness or rigor gap in the proof, but it is not circularity: the missing moments are not assumed to equal the theorem's conclusion. For the same reason, the p>0 statement in Theorem 3.4 is a statement-strength issue rather than a circular reduction. Overall, the derivation chain is self-contained against its assumptions and does not reduce to its own inputs.

Assumptions & free parameters 1 free parameters · 8 assumptions · 0 invented entities

The paper introduces no new entities and fits no parameters to data. Its central theorems assume a joint one-sided Lipschitz condition, polynomial growth, and specific two-sided bounds on the adaptive step. The existence/uniqueness of the continuous SDE and the particle approximation rates are taken from prior work, including the authors' own [13, 26], with proofs omitted. These are auxiliary to the new discretization error bounds but mean the full result depends on external results.

free parameters (1)
  • Adaptive timestep constants Ch, α1, α2, β, ϖ = Example 4.1: Ch=1, α1=α2=1, β=2, ϖ=10
    Hand-chosen in the numerical examples to make the chosen h satisfy Assumption 3.2; they are not fitted to data and do not enter the general convergence theorems.
assumptions (8)
  • domain assumption One-sided Lipschitz/monotonicity condition on b, σ, σ0 jointly (Assumption 3.1(1))
    Used throughout to get moment bounds and the contraction in the error equations; if it fails the well-posedness and convergence arguments break.
  • domain assumption Polynomial growth condition on drift and diffusion (Assumption 3.1(2) and Remark 3.1)
    Used to bound the adaptive step and the local error terms in Theorems 3.1-3.4.
  • domain assumption Strong dissipativity in infinite horizon (Assumption 3.3, λ1 > λ2 + 5/2)
    Needed for the uniform-in-time moment bound and decay in Theorem 3.4.
  • domain assumption Existence/uniqueness of strong solution of (1.2) under Assumption 3.1 (Lemma 3.1)
    Cited from [21, Theorem 2.1]; the paper does not prove it.
  • domain assumption Two-sided bounds on the adaptive timestep function (Assumption 3.2 and 3.4)
    The whole scheme is defined by such h; the rates hold only for step choices satisfying these bounds, including the lower bound (3.5).
  • domain assumption Propagation of chaos rates for the continuous particle system (Proposition 3.2)
    Quoted from [13, Theorem 3.2] with proof omitted; used to connect the particle system to the mean-field limit, but not to the discretization error.
  • domain assumption Exponential decay propagation of chaos (Proposition 3.3)
    Combined from [26] and [35] with proof omitted; supports the infinite-horizon particle approximation.
  • standard math Standard stochastic calculus (Itô formula, Gronwall inequality, Wasserstein metric properties)
    Used implicitly in all proofs.

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Pith. "Pith review of The adaptive EM schemes for McKean-Vlasov SDEs with common noise in finite and infinite horizons." pith.science (2026). https://pith.science/paper/WMILPNXM

@misc{pith2026250900521,
  author       = {Pith},
  title        = {Pith review of: The adaptive EM schemes for McKean-Vlasov SDEs with common noise in finite and infinite horizons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WMILPNXM}},
  note         = {Machine review of arXiv:2509.00521}
}
abstract

This paper is dedicated to investigating the adaptive Euler-Maruyama (EM) schemes for the approximation of McKean-Vlasov stochastic differential equations (SDEs) with common noise. When the drift and diffusion coefficients both satisfy the superlinear growth conditions, the $L^p$ convergence rates in finite and infinite horizons are revealed, which reacts to the particle number and step size. Subsequently, there is an illustration of the theory results by means of two numerical examples.

Discussion (0). Continue with ORCID to comment.

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