{"id":"d045ff57-416b-484e-afc7-1516dede2f30","arxiv_id":"2412.17418","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A convergence-rate analysis of Euler and particle discretizations for McKean-Vlasov SDEs with common noise under Lipschitz and Holder assumptions, with weaker regularity than prior Milstein-type schemes.","lead":"Le Gall proves convergence rates for a two-step Euler-plus-particles numerical scheme for McKean-Vlasov stochastic differential equations that include a common, market-wide noise. The work gives explicit error bounds separating time-discretization error from particle-approximation error, with an application to interbank lending models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.3's proof swaps the non-interacting empirical measure ν^N for the interacting μbar^N in (4.4), and Theorem 1.2 only supplies a grid-time sup, so the main sup_t bound does not follow as written.","rationale":"The reader's weakest-assumption captures a real gap in Theorem 1.2(ii): Lemma 4.3 is applied pointwise in ω0 and in t, and the paper does not explicitly justify passing to sup_t and to an L^p bound with an integrable constant. That gap is fixable: for q = p+ε, sup_t M_q(μbar_t(ω0))^{p/q} is dominated by (E1[sup_t|Y^1_t|^q])^{p/q}, whose L^1 norm is finite by Lemma 3.2, so dominated convergence repairs the step. However, the more immediate blocker for the central claim is in the proof of Theorem 1.3 itself. The paper writes that (4.4) gives a bound in terms of sup_t ||W_p(μbar^N_t, μbar_t)||_p, but (4.4) is explicitly stated with g(t) involving W_p(μbar_t, ν^N_t), the non-interacting empirical measure. Moreover, Theorem 1.2 is only proved for sup over grid points m ∈ {1,...,M}, while (4.8) in Theorem 1.3 requires sup over all t ∈ [0,T]. Neither issue is fatal: replacing μbar^N by ν^N in the application of (4.4) and using the intermediate sup_t bound on W_p(μbar_t, ν^N_t) from the proof of Theorem 1.2(ii) would close the argument, after the uniform-in-time constant is properly integrated. The Euler rate of Proposition 3.1 is carefully derived and the numerical experiments are consistent with the stated rates, so the paper's core approach appears sound but the proof of the headline bound needs correction. The reader's conditional verdict is therefore appropriate; no verdict change is needed.","tokens_in":23511,"tokens_out":12861,"duration_ms":108831,"concrete_test":"Re-derive Theorem 1.3's argument starting from (4.3)–(4.4) and track which empirical measure appears in g(t). If g(t) uses W_p(μbar_t, ν^N_t), then check whether the proof of Theorem 1.2 establishes sup_{t∈[0,T]} ||W_p(μbar_t, ν^N_t)||_p ≤ C E_N with an integrable constant; for the uniform-in-time step, test whether sup_t M_q(μbar_t(ω0))^{p/q} has finite L^1 norm via Lemma 3.2 and the estimate sup_t E1|Y^1_t|^p ≤ E1[sup_t|Y^1_t|^p]. Then verify whether (4.8) is proved for continuous t; if only the grid version exists, one must either add temporal-regularity control for W_p(μbar^N_t, μbar_t) or replace μbar^N by ν^N throughout the final step. Recomputing the line 'By the inequality (4.4)' with the actual definition of g(t) immediately settles whether the swap is a typo or a genuine gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1.3, after the Minkowski split (4.7), the text invokes '(4.4)' to claim sup_t ||X^{1,N}_t - Y^1_t||_p ≤ C sup_t ||W_p(μbar^N_t, μbar_t)||_p. But inequality (4.4) is derived from (4.3) with g(t) defined through W_p(μbar_r, ν^N_r), where ν^N is the empirical measure of the non-interacting copies Y^i of the Euler scheme, not the interacting particle empirical measure μbar^N. The right-hand side should involve W_p(μbar_t, ν^N_t), not W_p(μbar^N_t, μbar_t). Separately, Theorem 1.2 is stated and proved for sup_{m∈{1,...,M}} at grid times; the proof of Theorem 1.3 then uses as (4.8) the continuous-time statement sup_{t∈[0,T]} ||W_p(μbar^N_t, μbar_t)||_p ≤ C E_N, which is neither stated nor derived. Combining (4.4) with the grid version yields only sup_m for the interacting empirical measure, through (4.5). To close Theorem 1.3 one must first establish a sup_t bound for W_p(μbar_t, ν^N_t) (attempted in the proof of Theorem 1.2(ii), with an unproved uniform-in-time/integrability step for the Fournier–Guillin constant), then transfer via Gronwall to X^{1,N}-Y^1 and finally add the Euler error of Proposition 3.1. As written, this chain has a missing continuous-time estimate and an apparent substitution of one empirical measure for another.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes and analyzes a numerical scheme for McKean-Vlasov stochastic differential equations with common noise. The scheme first discretizes time by an Euler scheme for the conditional-law SDE (1.4)-(1.5) and then approximates the conditional law by the empirical measure of N interacting particles (1.6)-(1.7). The main results are Proposition 3.1, giving an Euler error O(h^{1/2 ∧ ρ}) under Lipschitz and Hölder assumptions; Theorem 1.2, giving rates for the Wasserstein distance between the particle empirical measure and the conditional law; and Theorem 1.3, combining these into a strong error bound of order h^{1/2 ∧ ρ} + E_N for the particle method. Two numerical examples, a conditional Ornstein-Uhlenbeck process and an interbank market model, are provided along with simulation code.","tokens_in":23892,"tokens_out":15727,"duration_ms":126864,"significance":"If valid, Theorem 1.3 gives a clean separation of the total strong error into a time-discretization error and an empirical-measure (particle) error for McKean-Vlasov equations with common noise, extending the framework of Liu [24] to the conditional-law setting. The rates are derived from external benchmarks (Carmona-Delarue conditional-law results, Fournier-Guillin empirical measure bounds, and standard BDG/Gronwall inequalities) rather than fitted to the target quantity, and the paper carefully contrasts its assumptions with the Milstein-type scheme of Biswas et al. [4]. The proof strategy is natural and the claimed rates are plausible. However, as detailed below, two steps in the proof of the particle error are not justified as written, so the central claims require revision rather than immediate acceptance.","major_comments":[{"comment":"The proof invokes inequality (4.4) to claim that ||sup_t |Xbar^{1,N}_t - Ybar^1_t|||_p is bounded by C sup_t ||Wp(μbar^N_t, μbar_t)||_p, but inequality (4.4) in the proof of Theorem 1.2 bounds this quantity by a constant times g(t), where g is defined through sup_r Wp(μbar_r, ν^N_r) with ν^N the empirical measure of the non-interacting particles Y^i, not through the interacting empirical measure μbar^N. Consequently the displayed chain leading to (4.8) and the final bound (1.9) is not justified as written. The proof can be repaired by using the rate for sup_t ||Wp(μbar_t, ν^N_t)||_p obtained in the proof of Theorem 1.2(ii) and by deriving a continuous-time version of (4.5), but as it stands the central estimate of Theorem 1.3 does not follow.","section":"Section 4, proof of Theorem 1.3 (after Eq. (4.7))"},{"comment":"The paper asserts that for every ω0 ∈ Ω0, sup_{0≤t≤T} E1[W_p^p(μbar_t(ω0), ν^N_t(ω0))] ≤ C(...) with C independent of N. Lemma 4.3 provides a bound whose constant is C M_q^{p/q}(μbar_t(ω0)) with q=p+ε, so the constant in the supremum-in-t bound is A(ω0)=C sup_t M_q^{p/q}(μbar_t(ω0)), which depends on ω0. Passing to the Lp(P0) rate for sup_t ||Wp(μbar_t, ν^N_t)||_p by the dominated convergence theorem requires integrability of A (or a similar domination). This is not proved in the text; it can be derived from Lemma 3.2 because A(ω0) ≤ (E1[sup_t |Ybar^1_t(ω0,·)|^q])^{p/q}, but the missing argument means the sup_t rate in Theorem 1.2(ii) is not fully established as written.","section":"Section 4, proof of Theorem 1.2(ii) (application of Lemma 4.3)"},{"comment":"Theorem 1.2 is stated for sup_{m∈{1,...,M}} at grid times, yet the proof's final display and the estimate (4.8) in Theorem 1.3 use continuous-time suprema such as sup_{t∈[0,T]} ||Wp(μbar_t, ν^N_t)||_p and sup_{t∈[0,T]} ||Wp(μbar^N_t, μbar_t)||_p. The inequality (4.5) controls only the grid-time supremum, and the continuous-time extension is asserted without proof. Since Theorem 1.3 depends on this extension, it should be stated and proved explicitly.","section":"Section 4, proof of Theorem 1.2 (final display) and Theorem 1.3 (Eq. (4.8))"}],"minor_comments":[{"comment":"The same symbol s is used for the continuous time and for the grid point floor(s); adopting distinct notation (e.g., \\underline{s}) would remove ambiguity.","section":"Section 3, proof of Proposition 3.1 (Eqs. (3.3)-(3.6))"},{"comment":"The third-case condition 'p ∈ (0, d/2)' is inconsistent with the standing assumption p ≥ 2; the intended condition is d > 2p.","section":"Section 1.3, Theorem 1.2"},{"comment":"The text states that the volatility with respect to the local shock is σρ and with respect to the global shock is σ√(1-ρ²), but equation (5.5) has σ√(1-ρ²) dW^i_t and σρ dW^0_t; the verbal description is reversed and should be corrected.","section":"Section 5.2, Interbank market model"},{"comment":"Reference [20] is missing publication details (publisher and year).","section":"References"},{"comment":"The figure captions report empirical slopes but do not discuss their relation to the theoretical rates h^{1/2∧ρ} and E_N; a brief comment would help the reader assess the simulations.","section":"Section 5, figures"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's core idea and claimed rates are sensible, and the gaps identified above appear patchable within the scope of a revision rather than requiring a new approach. The main worry is the misquotation of (4.4) in the proof of Theorem 1.3, which is a concrete error in the central argument; the authors should rewrite the proof of Theorem 1.3 using the correct empirical measure (ν^N) and supply the missing uniformity/integrability argument in Theorem 1.2(ii). The simulations are illustrative but not a quantitative validation of the rates."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine extension of the Liu Euler-particle analysis to common noise, with a clean Euler rate and a particle rate that is probably correct but whose written proof has a real gap. Theorem 1.3 as written does not follow.\n\nWhat is new: an Euler scheme that keeps the measure argument, Lipschitz coefficients only, p>=2, and rate h^{min(1/2,rho)}. That covers cases like sigma(x)=|x| which the Milstein-type scheme in [4] excludes. Proposition 3.1 is solid: standard Gronwall/BDG with conditional measures, no hidden circularity. The comparison with [4] is accurate, and the numerical examples are illustrative but honest. The conditional-law setup follows Carmona-Delarue correctly.\n\nThe soft spot is in Section 4. Theorem 1.2(ii) is supposed to give a sup_t rate. The proof applies Fournier-Guillin pointwise in t and then jumps to sup_t with an integrable constant; the uniform-in-t step and the integrability of the constant are not proved. That looks fixable, but it is not in the paper. More seriously, Theorem 1.3 claims sup_t ||Xbar^{1,N}-Y^1||_p <= C sup_t W_p(mubar^N_t, mubar_t) from (4.4). But (4.4) is fed by g(t) built from W_p(mubar_r, nu^N_r), where nu^N is the empirical measure of the non-interacting Euler copies, not from the interacting mubar^N. The right-hand side should involve sup_t W_p(mubar_t, nu^N_t). Then (4.8) claims a continuous-time sup_t bound for mubar^N vs mubar from Theorem 1.2, but Theorem 1.2 states only a grid sup. So the main sup_t bound is not established as written. This is a load-bearing gap in the proof of the headline particle rate, though the ingredients are very close: (4.4) plus a genuine sup_t bound on W_p(mubar, nu^N) would close Theorem 1.3. I would not call the result false; I would call the proof incomplete.\n\nBottom line: worth a serious referee. The Euler result alone justifies the paper. The particle theorem needs a patch before publication. A careful referee should ask for a rewritten Section 4 with a real sup_t Fournier-Guillin estimate and the corrected empirical measure in (4.4)/(4.8). Anyone using this should treat the particle rate as conditional on that fix.","headline":"Clean Euler rate for common-noise McKean-Vlasov, but the headline particle sup_t rate has a proof gap in Section 4; likely fixable, needs revision before I'd trust the main theorem.","tokens_in":24436,"tokens_out":4639,"would_cite":true,"duration_ms":40969,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","65C30","65C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"A common-noise mean-field SDE can be simulated by a two-stage Euler-plus-particle scheme whose strong error is $C(h^{1/2\\wedge\\rho}+E_N)$.","keywords":["McKean–Vlasov equation","common noise","particle method","Euler scheme","propagation of chaos","Wasserstein distance","empirical measure","mean field games"],"falsifier":"Construct a common-noise coefficient whose conditional law has a $(p+\\varepsilon)$-moment that grows without bound along some common-noise path, run the particle method (1.6)–(1.7) with fixed time step and increasing $N$, and measure $\\sup_{t\\in[0,T]}W_p(\\bar{\\mu}^N_t,\\bar{\\mu}_t)$; if this supremum decays slower than $N^{-1/(2p)}$ (or the appropriate dimension-dependent rate), the uniform-in-time step in Theorem 1.2(ii) fails.","tokens_in":23316,"feed_emoji":"🎲","tokens_out":9811,"duration_ms":88015,"temperature":0.7,"pith_summary":"This paper proves that the McKean–Vlasov equation with common noise can be simulated by freezing the coefficients on an Euler grid and then replacing the conditional law in the coefficients by the empirical measure of $N$ particles, and that the combined strong error is bounded by a time-discretisation term $h^{1/2\\wedge\\rho}$ plus a particle term $E_N$ that decays like $N^{-1/(2p)}$, $N^{-1/d}$, or a log-corrected version depending on the dimension. The point is that adding a common noise — a Brownian driver shared by all particles and affecting the coefficients through the conditional law — does not destroy the clean error separation known for the standard McKean–Vlasov case. If the bound is right, users can choose the number of particles and the time step to balance the two error sources, and the method applies to coefficients, such as $|x|$ or a square-root-like diffusion, that schemes requiring differentiability of the diffusion cannot handle.","feed_headline":"Common-noise mean-field simulation now comes with proven error rates","feed_subtitle":"A two-stage scheme splits the error cleanly into a time-step term and a finite-particle term.","key_machinery":"The auxiliary non-interacting particle system (4.1): $N$ conditionally i.i.d. copies $\\bar{Y}^i$ of the continuous Euler scheme, all driven by the same common noise $W^0$. These copies play the role of the ideal particle cloud: by Lemma 4.1 their empirical measure $\\nu^N$ has the same conditional law as the Euler scheme's law, so classical empirical-measure estimates apply directly; the interacting particle system is shown to stay within a constant multiple of $\\nu^N$'s Wasserstein error, and the Euler scheme is shown to stay within $h^{1/2\\wedge\\rho}$ of the true solution.","core_discovery":"The central result is Theorem 1.3: for $p\\in[2,\\infty)$, under Lipschitz continuity in the state and measure arguments and $\\rho$-Hölder continuity in time, the first particle path $\\bar{X}^{1,N}$ driven by the same Brownian motions as the true solution $X$ satisfies $\\|\\sup_{t\\in[0,T]}|\\bar{X}^{1,N}_t-X_t|\\|_p \\le C(h^{1/2\\wedge\\rho}+E_N)$. The Euler half of the rate is Proposition 3.1, and the particle half is Theorem 1.2(ii): at the grid times, the $p$-Wasserstein distance between the particle empirical measure $\\bar{\\mu}^N_{t_m}$ and the conditional law $\\bar{\\mu}_{t_m}$ decays at the classical empirical-measure rate, with an exponent that depends on whether $p$ is larger, equal, or smaller than $d/2$. The proof route is to compare the interacting particles with $N$ conditionally i.i.d. copies of the continuous Euler scheme, transfer the empirical-measure error to the interacting system through a Gronwall-type estimate, and then add the Euler error.","pith_inferences":["A direct check of the uniformity step in Theorem 1.2(ii) would compute $\\sup_{0\\le t\\le T} E_1[W_p^p(\\bar{\\mu}_t(\\omega^0),\\nu^N_t(\\omega^0))]$ for a non-Gaussian example and compare the empirical slope with $E_N^p$; if the slope is worse, the uniform-in-time step, not the single-time rate, is the fragile part.","Because the paper's scheme freezes the measure argument at grid points, replacing the empirical measure by a quantization grid, as already done for the no-common-noise version, should inherit the same $h^{1/2\\wedge\\rho}$ time error; this is an extension the author points toward but does not prove.","The result is a strong error in the shared-Brownian-motion coupling; it does not by itself give an $L^p$ bound on the distance between marginal laws, so density estimation or weak error would need a separate argument."],"forward_implications":["For a fixed computational budget, the error bound shows how to balance the time step and the particle number: once $h^{1/2\\wedge\\rho}$ is comparable to $E_N$, refining time without adding particles no longer reduces the leading error.","The method requires no differentiability of the diffusion coefficients, so non-smooth maps such as $x\\mapsto|x|$ or a square-root-like diffusion are admissible under the stated assumptions.","Because the Euler discretisation keeps the measure argument symbolic, other spatial discretisations, such as quantization, can replace the empirical measure without redoing the time-error analysis.","In dimensions $d<2p$ the particle error decays only as $N^{-1/d}$, so the particle count dominates the total error once the time step is small; in higher dimensions the $N^{-1/(2p)}$ rate applies."],"supporting_citations":[{"why":"Supplies the conditional-law formalism, the continuity of the conditional-law process, and the strong well-posedness of the McKean–Vlasov SDE with common noise.","marker":"[9]"},{"why":"Introduces the Euler-plus-particle construction for McKean–Vlasov equations that this paper extends to common noise.","marker":"[6]"},{"why":"Provides the particle method with the empirical-measure estimator and the corresponding rate analysis for the case without common noise, the template for Theorems 1.2 and 1.3.","marker":"[24]"},{"why":"Supplies the almost-sure convergence of empirical measures used to pass from pointwise to uniform convergence in Theorem 1.2(i).","marker":"[20]"},{"why":"Supplies the quantitative Wasserstein rate for empirical measures, which enters as $E_N$ in both Theorem 1.2(ii) and Theorem 1.3.","marker":"[15]"}],"fun_headline_variants":["Proven error bounds for common-noise mean-field particle methods","Two-stage error split for common-noise McKean-Vlasov","Euler-plus-particles error bounds for common noise","Sharp error rates for common-noise McKean-Vlasov simulation","Common-noise simulation gets proven convergence rates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the conditional empirical measure $\\nu^N$ approximates the Euler scheme's conditional law uniformly over the whole time interval at the same rate as at a single time, with an error constant that stays integrable in the common-noise randomness.","fun_headline_variants_meta":{"raw":{"variants":["Proven error bounds for common-noise mean-field particle methods","Two-stage error split for common-noise McKean-Vlasov","Euler-plus-particles error bounds for common noise","Sharp error rates for common-noise McKean-Vlasov simulation","Common-noise simulation gets proven convergence rates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000653,"raw_usage":{"total_tokens":2973,"prompt_tokens":906,"completion_tokens":2067,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":522,"completion_tokens_details":{"reasoning_tokens":1987}},"tokens_in":522,"tokens_out":2067,"duration_ms":14188,"temperature":1.0,"reasoning_tokens":1987,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:27:54.477332+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a common-noise coefficient whose conditional law has a $(p+\\varepsilon)$-moment that grows without bound along some common-noise path, run the particle method (1.6)–(1.7) with fixed time step and increasing $N$, and measure $\\sup_{t\\in[0,T]}W_p(\\bar{\\mu}^N_t,\\bar{\\mu}_t)$; if this supremum decays slower than $N^{-1/(2p)}$ (or the appropriate dimension-dependent rate), the uniform-in-time step in Theorem 1.2(ii) fails.","supporting_citations":[{"cited_title":"Carmona and F","cited_arxiv_id":null,"evidence_quote":"Supplies the conditional-law formalism, the continuity of the conditional-law process, and the strong well-posedness of the McKean–Vlasov SDE with common noise."},{"cited_title":"A stochastic particle method for the McKean-Vlasov and the Burgers equation","cited_arxiv_id":null,"evidence_quote":"Introduces the Euler-plus-particle construction for McKean–Vlasov equations that this paper extends to common noise."},{"cited_title":"Particle method and quantization-based schemes for the simulation of the McKean-Vlasov equation","cited_arxiv_id":null,"evidence_quote":"Provides the particle method with the empirical-measure estimator and the corresponding rate analysis for the case without common noise, the template for Theorems 1.2 and 1.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the almost-sure convergence of empirical measures used to pass from pointwise to uniform convergence in Theorem 1.2(i)."},{"cited_title":"On the rate of convergence in Wasserstein distance of the empirical measure","cited_arxiv_id":null,"evidence_quote":"Supplies the quantitative Wasserstein rate for empirical measures, which enters as $E_N$ in both Theorem 1.2(ii) and Theorem 1.3."}],"review_version":1}