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The paper proves explicit combined particle-number and time-step error rates, and a computable almost-sure exponential stability rate, for the truncated Euler–Maruyama scheme applied to superlinear McKean–Vlasov proportional-delay jump mode

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2026-08-01 20:57 UTC pith:UXXZ3FCH

load-bearing objection Plausible extension of truncated Euler–Maruyama to proportional-delay McKean–Vlasov with jumps, but Theorem 4.1's key estimate skips delayed terms and the stability theorems are underproved; worth a serious referee. the 3 major comments →

arxiv 2607.16438 v1 pith:UXXZ3FCH submitted 2026-07-17 math.NA cs.NA

Convergence and stability of truncated Euler-Maruyama algorithm for stochastic proportional delay Mckean-Vlasov models with jump process

Amr Abosenna , Zhuoqi Liu This is my paper
classification math.NA cs.NA MSC 65C3060H3560H10
keywords McKean-Vlasov equationsproportional delayLévy jumptruncated Euler-Maruyamaconvergence ratealmost sure exponential stabilityinteracting particle systempropagation of chaos
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to give a quantitative error bound and a long-run stability guarantee for a practical numerical method applied to McKean–Vlasov stochastic differential equations with proportional (pantograph) delay and Lévy jumps, where the drift and diffusion coefficients may grow superlinearly. The scheme is the truncated Euler–Maruyama method driven by an interacting N-particle system, with the law replaced by the empirical measure. The main convergence theorem splits the total error into a particle-number part coming from propagation of chaos (of order N^{-1/2} or N^{-p/d} depending on dimension) and a time-step part that, for polynomial growth ε, behaves like a power of Δt and approaches Δt^{1/(1+ε)} when a technical parameter is large. A second theorem gives almost sure exponential stability of the averaged squared numerical solution, with rate γ=min(1,ς*) computed from the coefficients. If correct, the paper supplies users of such simulations with explicit trade-offs between number of particles, step size, and guaranteed accuracy, plus a stability threshold.

Core claim

The central claim is that for a McKean–Vlasov SDE with pantograph delay and Lévy jump, under one-sided Lipschitz/dissipativity conditions on the drift and diffusion and a jump integrability condition, the truncated Euler–Maruyama particle scheme converges in L^p uniformly in time with error O(N^{-1/2} + Δt^r), where the N-rate becomes N^{-p/d} or N^{-1/2} log(1+N) in low or high dimension, and the Δt exponent r is explicitly [γ(q−(1+ε)p)/(1+ε)] ∧ [(q−εp)/q] with λ(r)=C r^{1+ε}, κ(Δt)=Δt^{-γ}, and q between q0 and q*. The same scheme is almost surely exponentially stable in the averaged squared norm, with rate min(1,ς*) and ς* the unique positive root of Eq. (92). The paper's message is that

What carries the argument

The load-bearing device is the truncation map in Eqs. (18)–(19): f and g are evaluated at arguments projected to a ball of radius λ^{-1}(κ(Δt)); with λ(r)=C r^{1+ε} and κ(Δt)=Δt^{-γ} this gives bounded coefficients of size roughly Δt^{-γ} and an explicit truncation radius. That single map is what makes superlinear coefficients tractable, and the explicit choice of λ and κ is what converts the generic convergence estimate into a readable power of Δt in the final rate. The second load-bearing object is the empirical measure of the N-particle system, which replaces the law υ_t by the average of the N particles; the propagation-of-chaos estimate (Proposition 2.1) supplies the N-dependent part of

Load-bearing premise

The convergence-rate theorem depends on Assumption 2.2 being strong enough that the drift and diffusion difference between the exact and numerical errors closes with a single Gronwall inequality; as written, the proof of the J1 term in Eq. (66) silently drops delayed-error and Wasserstein terms that would need a separate sup-type argument.

What would settle it

Take a one-dimensional model satisfying the assumptions with ε=2, q0=5, q*=6, η=0.5, run the truncated scheme with λ(r)=r^3, κ(Δt)=Δt^{-γ}, γ=1/6, and compare L^2 errors at Δt=2^{-8},...,2^{-12} with N large; the predicted exponent is [γ(q−(1+ε)p)/(1+ε)]∧[(q−εp)/q] with p=2, q=5.9. If the observed slope is worse than the formula, the dropped terms in the J1 estimate are contributing an extra time-step factor. For the same test, increase N to isolate the Δt rate and check whether it approaches 1/(1+ε)=1/3 as q* grows; a slope stuck well below 1/3 would falsify Remark 4.1.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • A user can balance N and Δt: increasing the number of particles improves the N^{-1/2} (or N^{-p/d}) part independently of the Δt part, so the optimal allocation for a target error is set by the two exponents.
  • For polynomial growth ε, taking q large pushes the time-step exponent toward 1/(1+ε), which the paper calls the optimal rate for the jump case; without jumps the analogous rate is close to p/2.
  • The stability theorem gives an explicit threshold (c1−c3)>(c2+c4)(1+[1/η]) and a computable decay rate γ=min(1,ς*) for the averaged squared numerical solution, so one can check before simulating whether long-run decay is guaranteed.
  • The exact interacting particle solution is also almost surely exponentially stable, with rate min(1,(c1−c3)−(c2−c4)), so the numerical stability claim is matched by a statement about the underlying particle system.
  • Numerical examples with λ(r)=r^5 and r^6 and κ=Δt^{-1/10} show empirical root-mean-square error slopes of about 1/2, consistent with the predicted order when the Δt term is not dominant.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the claimed optimal Δt^{1/(1+ε)} rate rests on letting q*→∞; for finite q*, the proved bound has the worse exponent [γ(q−(1+ε)p)/(1+ε)]∧[(q−εp)/q], so a testable prediction is that the observed convergence slope declines as q0/q* approaches 1.
  • Editorial extension: if the missing J1 Gronwall step in Eq. (66) is supplied, one likely needs a sup-type Gronwall inequality, which would typically introduce at most a logarithmic factor in Δt; the paper's displayed rates might then be off by a log factor in the worst case.
  • Editorial extension: because the jump coefficient h is not truncated, the same framework suggests the natural next problem of heavy-tailed Lévy measures where only moments up to order 2+δ exist, and whether the p/2 exponent in the jump part weakens.
  • Editorial extension: in a multilevel Monte Carlo setting, the explicit combined rate could be used to choose pairing of N and Δt across levels to minimise computational cost for a target mean-square error.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper studies a McKean–Vlasov stochastic differential equation with proportional delay and Lévy jumps, where the drift and diffusion may grow superlinearly. A truncated Euler–Maruyama scheme is applied to an interacting particle system. The paper claims an L^p convergence rate in both the number of particles N and the time step Δt (Theorem 4.1, Corollary 4.2), and almost sure exponential stability of the numerical solution (Theorem 5.2). Four numerical examples are provided.

Significance. If correct, the paper would give a combined particle-number/time-step convergence rate for a truncated EM scheme in a fairly general superlinear McKean–Vlasov jump-delay setting, together with a long-time stability result. The explicit rates, the use of a truncation function with κ(Δt)=Δt^{-γ}, and the numerical validation are strengths. However, the main convergence theorem currently has a gap in the treatment of delayed and mean-field error terms, and the jump coefficient is not assumed to satisfy the L^2(π) growth conditions used in the proofs. These issues are repairable but affect the central claim.

major comments (3)
  1. [§4, proof of Theorem 4.1, Eq. (66)] The bound J1 ≤ C∫_0^t E|e_i(s∧τ_i)|^p ds is not a direct consequence of Assumption 2.2. Applying Assumption 2.2 inside J1 gives additional terms |x^{i,N}_{ηs}−ϖ^{i,N}_{ηs}|^2, W_2^2(υ^{x,N}_s,υ^{ϖ,N}_s), and W_2^2(υ^{x,N}_{ηs},υ^{ϖ,N}_{ηs}) under the |e_i(s)|^{p−2} factor. The Wasserstein terms can be bounded by averages of |e_j(s)|^2, but the delayed term is not controlled by E|e_i(s)|^p; a sup-type Gronwall argument over sup_{0≤r≤s}E|e_i(r)|^p is required and absent. Since Eq. (76) and the rates (53)/(57) depend on (66), the central convergence claim is not established as written. The unlabelled passage from t∧τ_i to t in (76) also needs a localization argument.
  2. [Assumption 2.1 and Lemma 4.2 / J3] Assumption 2.1 is insufficient for the jump estimates used in Lemmas 4.1–4.2 and J3 of Theorem 4.1. The proofs bound E∫|h|^2π(dz)ds and E∫|h|^pπ(dz)ds by moments of the numerical solution (e.g., Eq. (30), Eq. (43), Eq. (75)). However, Assumption 2.1 only gives an L^1(π) Lipschitz/linear-growth condition on h (Eqs. (3),(9)); because π is finite, L^1(π) does not imply L^2(π), and ∫|z|^pπ(dz) is not assumed for p>2. A global L^2(π)-growth condition on h, together with the needed p-th moment condition, must be added or derived; otherwise Lemma 4.2—and hence the moment bounds used throughout §4—is not justified.
  3. [Remark 4.1, Eq. (82)] The claimed optimal rate Δt^{1/(1+ε)} in Eq. (82) is obtained by letting q→∞ after setting γ=1/q. Corollary 4.2 requires q∈((1+ε)p∨q_0,q*), and Assumption 2.3 only asserts q*>q_0. Thus the passage q→∞ is not permitted under the stated assumptions. The remark should be formulated conditionally, or an additional assumption that q* can be taken arbitrarily large should be stated.
minor comments (6)
  1. [Lemma 4.2, Eq. (35)] The Itô formula displayed in Eq. (35) evaluates several integrands at t while integrating with respect to ds, dW(s), or Ñ(ds,dz); these should be s. This makes the proof difficult to follow and should be corrected.
  2. [Theorem 5.2, Eq. (110)] The statement “lim sup ≤ ∞” is vacuous; it should be “lim sup e^{γlΔt} (1/N)Σ|ϖ_l|^2 < ∞ a.s.” Also, Definition 5.2 uses log of the average of |ϖ_l|, while Theorem 5.2 proves a bound for the average of |ϖ_l|^2; the definitions should be aligned.
  3. [Corollary 4.1, Eq. (33)] The proof says “the same approach as Lemma 4.1,” but for general t, ηt need not lie in the same grid interval as t; the case [ηt] ≠ [ηl] requires an additional one-step estimate. Please spell this out.
  4. [Theorem 5.1] The proof of Theorem 5.1 is delegated to [46]. Since the present model includes jumps, delay, and McKean–Vlasov interaction, the authors should either provide a proof or state precisely how the arguments of [46] adapt.
  5. [Example 6.3] The text chooses λ(r)=r^7, which corresponds to ε=6, but then states ε=4 and reports rate 0.16 = 1/(1+4)−2/50. Please reconcile the parameters.
  6. [Example 6.4] The constants are listed as “c_1=12, c_2, c_3=2 and c_4=1”; the value of c_2 appears to be missing.

Circularity Check

0 steps flagged

No load-bearing circularity: central convergence/stability derivations are conditional on explicit coefficient assumptions, and the lone self-citation (Ref. [45]) supplies only the standard stability definitions.

full rationale

The paper's main results (Theorem 4.1/Corollary 4.2 and Theorem 5.2) are derived from Assumptions 2.1–2.3, 5.1, 5.2, the truncation framework of Mao [37], and the externally cited propagation-of-chaos bound [41]; none of these are refitted to the target errors. No quantity is defined in terms of the quantity it is said to predict, and no fitted parameter is renamed as a prediction. The only self-citation is Ref. [45] (Liu, Gao, Yuan, Guo), used in Definitions 5.1 and 5.2 for the meaning of almost sure exponential stability; these definitions are not the derived conclusion and do not carry the proof. The proof of Lemma 2.1 refers to [40] and Theorem 5.1 to [46], both external. The numerical examples compare with very-small-step reference solutions, which is external sanity checking rather than circular. One proof gap should be distinguished from circularity: in Theorem 4.1, the transition to Eq. (66) claims J1 ≤ C∫E|e_i(s∧τ_i)|^p ds "directly" from Assumption 2.2, whereas Assumption 2.2 also yields delayed-error and Wasserstein terms; controlling them would require an extra sup-type Gronwall argument. That is an omitted/incorrect estimate and a correctness risk, not a reduction by construction or a fit-to-target, so it does not raise the circularity score. Remark 4.1's optimal-rate claim 1/(1+ε) invokes q→∞ and assumes q* can be arbitrarily large; again a hypothesis-strengthening concern, not circularity. Overall: no significant circularity.

Axiom & Free-Parameter Ledger

3 free parameters · 7 axioms · 0 invented entities

The paper introduces no new physical entities. Its contribution is a numerical-analysis theorem resting on coefficient assumptions, imported propagation-of-chaos estimates, and a user-chosen truncation schedule. The main hidden cost is the extra q*→∞ premise in the optimal-rate remark and the implicit sup-Gronwall closure in the convergence proof.

free parameters (3)
  • moment parameter q = chosen satisfying q > (1+ε)p∨q0 and q<q* (e.g., q=50 in Example 6.3)
    The convergence-rate exponent depends on q. Remark 4.1 takes q→∞ to claim the near-optimal rate 1/(1+ε), which requires q*→∞, an extra condition not in Assumption 2.3.
  • truncation exponent γ = κ(∆t)=∆t^{-γ}, γ∈(0,1/4∧1/q]; examples use γ=1/10, 1/50, 1/4
    The choice of γ controls the truncation level and appears directly in the claimed convergence-rate exponent. It is chosen by the authors, not derived.
  • superlinear growth parameter ε = ε≥0 in Assumption 2.1; examples take ε=4
    ε governs the claimed rate 1/(1+ε). It is a model constant rather than a fitted number, but the central rate claim is highly sensitive to it.
axioms (7)
  • domain assumption Assumption 2.1: f,g satisfy polynomial-weighted Lipschitz and Wasserstein continuity; h satisfies an integral Lipschitz condition with finite square-integrability against π.
    This is the main regularity condition on the coefficients and is assumed without proof.
  • domain assumption Assumption 2.2: one-sided dissipativity of f and g with q0>2 plus Wasserstein terms.
    Used critically in the convergence proof and in propagation of chaos; not derived.
  • domain assumption Assumption 2.3: one-sided growth condition with q*>q0.
    Provides the moment bounds for the solution and numerical solution. The optimal-rate claim needs q* large, which is not guaranteed by this assumption.
  • domain assumption Assumptions 5.1 and 5.2: dissipativity with constants c_i and quadratic bound on |f|² with b_i.
    These are the stability conditions used in Theorems 5.1 and 5.2; no evidence is given that they hold beyond the examples.
  • domain assumption Finite-activity Lévy measure: π(Z)<∞ and ∫|z|²π(dz)<∞.
    The proofs of the jump martingale estimates and the numerical scheme rely on finite activity and finite second moment.
  • standard math Lemma 2.1 (well-posedness) is imported from [40] and Proposition 2.1 (propagation of chaos) is imported from [41].
    Both are cited prior theorems. The paper provides no proofs and builds the main convergence theorem directly on them.
  • ad hoc to paper In Remark 4.1, q* is allowed to be arbitrarily large so that q→∞.
    This is not part of Assumption 2.3; it is invoked specifically to obtain the claimed optimal convergence rate 1/(1+ε).

pith-pipeline@v1.3.0-alltime-deepseek · 24412 in / 19976 out tokens · 170644 ms · 2026-08-01T20:57:26.022092+00:00 · methodology

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read the original abstract

Stochastic Mckean-Vlasov models have a substantial importance in different fields such as finance, biology and control. This paper puts the light on stochastic proportional delay Mckean-Vlasov model with L\'evy jump where the non-jump coefficients are granted the permission to grow beyond linearity. The truncated Euler-Maruyama algorithm is then applied to our addressed model where the convergence rate and almost sure exponential stability of the aforementioned numerical algorithm are being investigated. Finally, numerical examples are presented to foster the theoretical analysis done throughout the paper

Figures

Figures reproduced from arXiv: 2607.16438 by Amr Abosenna, Zhuoqi Liu.

Figure 1
Figure 1. Figure 1: Log-log plot of RMSE versus ∆t for Eq.(112). Example 6.2. Consider the following stochastic proportional delay Mckean￾Vlasov model with L´evy jumps dxt = (−2x 4 t + E[x 2 t ])dt + (x 5 t + xηt cos 2 (xt))dW(t) + Z Z (1 + xt)zNe(dt, dz), t ≥ 0 (115) with initial data x0 ∼ N(0, 1) where N(0, 1) is the standard normal distribution, η = 0.8 and compensator is given by π(dz)dt = f(z)dzdt where f(z) ∼ N(0, 1). I… view at source ↗
Figure 2
Figure 2. Figure 2: Log-log plot of RMSE versus ∆t for Eq.(120). Example 6.4. Consider the following stochastic proportional delay Mckean-Vlasov model with L´evy jumps dxt = (−x 3 t − x 5 t )dt + (xt cos 2 (xηt) − 0.5 Z Rd yν(dy))dW(t) + Z Z (0.4xt + 0.2xηt)zNe(dt, dz), t ≥ 0, (122) where x0 = 1, η = 0.5 and ν ∈ P(R d ). The compensator is given by π(dz)dt = 2f(z)dudt where f(z) ∼ N(0, 1). It can be also noticed that the coef… view at source ↗
Figure 3
Figure 3. Figure 3: Trajectories of the numerical solutions of Eq.(122) over interval [0, 2]. Funding: There are no funders to report. Conflicts of Interest: This work does not have any conflicts of interest. Data Availability Statement: Data sharing not applicable to this article. References [1] Clemens Guhlke, Paul Gajewski, Mario Maurelli, Peter K Friz, and Wolfgang Dreyer. Stochastic many-particle model for LFP electrodes… view at source ↗

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