{"id":"6fdc7c82-8e02-4c22-9e14-73328fe95b63","arxiv_id":"2505.14262","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A general four-condition patching argument gives time-uniform strong error bounds for SDDE numerical methods, applied to backward and truncated Euler-Maruyama methods and to invariant measures.","lead":"This paper proves a general framework under which numerical methods for stochastic delay differential equations have errors that stay bounded uniformly for all future time. The result applies to backward and truncated Euler-Maruyama schemes and supports long-time simulation and invariant-measure approximation.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The contraction step in Appendix A is algebraically invalid: the stated window length T does not imply (A.6), so the proof of Theorem 3.3 does not go through as written; the same defect appears in Appendix B.","rationale":"The paper's central claim is a time-uniform strong error bound obtained by patching finite-window estimates and exploiting exponential contraction of the true SDDE. The most load-bearing step is the contraction inequality (A.6) in Appendix A: it is the only place where the exponential decay is converted into a geometric factor that makes the infinite sum of window errors finite. That inequality is algebraically false as written, because it is equivalent to 2^p M2 <= 1/2 and the proof's chosen T is built from M1, not from the contraction constants. The same defect appears in Appendix B's derivation of Lemma 3.5, which is needed for Theorem 3.6 and hence for the segment-process convergence used in Theorem 3.7. I therefore cannot accept the proof as it stands, although the error looks repairable by choosing T appropriately. The reader's weakest-assumption field emphasized the exponential-contraction assumption itself rather than the incorrect use of it in the patching proof; since both concern the contraction mechanism but at different levels, I mark partial agreement. I also note that Lemma 5.7, the segment contraction verification for BEM, is stated without proof; that is a secondary gap in the application but does not replace the primary algebraic defect. My verdict is UNCHANGED relative to the reader's CONDITIONAL: the central idea is credible and the argument is likely fixable, but the stated theorems are not proved as written.","tokens_in":35924,"tokens_out":23648,"duration_ms":203182,"concrete_test":"Independently re-derive (A.4)-(A.7) under Condition 3.1 with explicit numbers, e.g. p=2, M2=1, M3=1, M1=1, tau=0.1. Compute T from the paper's definition and test whether (A.6) holds; it reduces to 4 <= 1/2 and fails. Then test the candidate repair T* = (1/M3) log(2^{p+1} M2) (or T* >= 4 tau with an analogous K1,K2 choice in Appendix B) and verify that the iterated recurrence (A.9) has a summable geometric factor with a finite constant independent of horizon. If the repaired choice works, the verdict stays CONDITIONAL; if no choice of T makes the factor <1 while keeping the finite-time constant C_T finite, the uniform-in-time claim is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"At (A.4)-(A.5), for t in [2T,3T] the proof bounds the error by 2^p M2 (sup_{[T-tau,T]} E|x-X|^p) e^{-M3 T} + K_T Delta^q. The window-patching argument requires this coefficient to be less than 1, and (A.6) asserts 2^p M2 e^{-M3 T} <= e^{-M3 T}/2. Dividing by e^{-M3 T} gives 2^p M2 <= 1/2, which is not implied by Condition 3.1. The proof defines T := 2 tau + (2 log(2M1))/M2, using the numerical moment bound M1 rather than the contraction coefficient M2 and rate M3; when M2 > 2^{-p} the asserted contraction factor cannot hold for any T. Appendix B repeats the pattern: T = 4 tau + (4 log(2K1))/K2 can be negative when K1 < 1/2, and the displayed equality 2K1 e^{-K2 T} = e^{-3 K2 T/4} is false unless tau=0. These are not cosmetic: this contraction is exactly what makes the accumulated error over successive windows summable in (A.9)-(A.11). A repair is plausible by choosing T large enough so that 2^p M2 e^{-M3 T} <= 1/2 (and similarly for K1,K2) and re-running the iteration, but as written the proof does not establish the claimed constant C independent of T in Theorem 3.3, nor the analogous Lemma 3.5 needed for Theorem 3.6.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a general technique for proving strong convergence over the infinite horizon for numerical methods for stochastic delay differential equations (SDDEs). The idea is to partition time into windows of fixed length and to compare the numerical solution on each window with a newly restarted true solution, using exponential contraction of the true SDDE to make the accumulated errors summable. Theorem 3.3 claims a uniform-in-time bound sup_{t\\ge -\\tau} E|x(t;0,\\xi)-X(t;0,\\xi)|^p \\le C\\Delta^q under Condition 3.1. Theorem 3.6 claims the analogous uniform bound for segment processes under Condition 3.4, and Theorem 3.7 uses this to show that the law of the numerical segment process converges to the invariant measure of the true segment process as k\\to\\infty and \\Delta\\to0. The technique is illustrated on the truncated Euler-Maruyama method and on the backward Euler-Maruyama method, with numerical experiments included.","tokens_in":36301,"tokens_out":9332,"duration_ms":91717,"significance":"If the proof gap identified below is repaired, the paper would make a useful contribution: it provides a framework for obtaining strong error bounds whose constants do not grow with the time horizon, and it gives a route to numerical approximation of invariant measures that avoids proving existence of an invariant measure for the numerical scheme itself. The paper correctly identifies exponential contraction of the true dynamics as the key structural assumption, and it applies the framework to two practically relevant methods. The reliance on finite-time convergence results and on explicit moment/contraction assumptions is clearly stated, so the overall strategy is transparent. However, the central window-patching argument in the appendices is currently not valid as written, and the omitted proof of Theorem 3.6 makes that part of the paper impossible to verify.","major_comments":[{"comment":"The inequality 2^p M2 e^{-M3 T} \\le e^{-M3 T}/2 is equivalent to 2^p M2 \\le 1/2, which is not implied by Condition 3.1. The proof defines T := 2\\tau + (2\\log(2M1))/M2, using the numerical moment bound M1 rather than the contraction constants M2 and M3; when M2 > 2^{-p}, no value of T satisfies the displayed inequality. Also, T can be nonpositive when M1 < 1/2, since M1 is only assumed positive. This step is load-bearing because the contraction coefficient is what makes the successive error terms summable in (A.9)-(A.11), so Theorem 3.3 is not established as written. A repair appears plausible by choosing T so that 2^p M2 e^{-M3 T} \\le 1/2 and then repeating the iteration, but the proof must be corrected explicitly.","section":"Appendix A, Eq. (A.6)"},{"comment":"The same algebraic defect occurs in the proof of Lemma 3.5. With T = 4\\tau + (4\\log(2K1))/K2, the displayed identity 2K1 e^{-K2 T} = e^{-3K2 T/4} is false unless \\tau = 0; the term 4\\tau introduces an extra factor e^{-K2\\tau}. Moreover, when K1 < 1/2, the quantity \\log(2K1) is negative and T may fail to be positive. Consequently the asserted bound 2(K1 e^{-K2 T} + C_T^2 \\Delta^q) \\le e^{-K2 T/2} is not justified. Since Lemma 3.5 is needed for Theorem 3.6, this is a second load-bearing gap. The repair is to choose T from K1 and K2 so that 2K1 e^{-K2 T} \\le 1/2 and to handle the \\Delta^q term separately for sufficiently small step size.","section":"Appendix B, Eq. (A.4)"},{"comment":"Condition 3.4 is stated for a generic moment order p in items (ii)-(iv), but Lemma 3.5 proves a second-moment bound and its proof invokes Condition 3.4 with power 2; Theorem 3.6 then concludes sup_k E\\|x_{t_k} - X_{t_k}\\|^2 \\le C\\Delta^q. As written, these results only follow when Condition 3.4 is specialized to p=2, or when the conclusion is stated for the same p as in the condition. In addition, the proof of Theorem 3.6 is omitted as 'similar' to Theorem 3.3; given that the patching argument in Lemma 3.5 needs repair, the paper should provide the details rather than rely on analogy.","section":"Section 3, Condition 3.4, Lemma 3.5, and Theorem 3.6"}],"minor_comments":[{"comment":"The condition list contains two items labelled (iii); the second occurrence should be labelled (iv).","section":"Section 3, Condition 3.1"},{"comment":"The constants b1, b2, b3 appear twice without any distinction, although the two sets are clearly meant to be different; one set should be barred or otherwise renamed.","section":"Section 4, Assumption 4.2"},{"comment":"The first displayed inequality in Appendix A writes E(x(t;0,\\xi)-X(t;0,\\xi)) without the absolute value and without the p-th power; it should be E|x(t;0,\\xi)-X(t;0,\\xi)|^p to match the rest of the proof.","section":"Appendix A, Eq. (A.1)"},{"comment":"The statement contains the typo 'satisyes' instead of 'satisfies'.","section":"Lemma 3.5"},{"comment":"The text says that C_{T2-T1} is a constant dependent on T2 - T2; this should be T2 - T1.","section":"Condition 3.4, item (iv)"},{"comment":"The notation z(t; T1, z_{T1}) is used without a prior definition; since z is a numerical scheme starting from time 0, the flow-property definition for restarting at T1 should be made explicit.","section":"Section 4, Lemma 4.9"}],"recommendation":"major_revision","confidential_remarks":"The core idea of the paper is plausible and the main gap is localized to the choice of the window length in Appendices A and B. I recommend major revision rather than rejection: the contraction step is fixable within the scope of the paper, but the proofs as written do not establish the headline theorems. The manuscript fits the journal's scope, and the distinction from the related weak-convergence work of Crisan et al. is legitimate. I would also ask the authors to supply the missing proof of Theorem 3.6 in the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read the paper and checked the stress-test note against the appendices. The note is right. The main theorem is not proven as written because the contraction factor is asserted from a T that cannot deliver it. In (A.4)-(A.6), the iteration needs 2^p M2 e^{-M3 T} ≤ e^{-M3T}/2, which divides to 2^p M2 ≤ 1/2. The chosen T = 2τ + (2 log(2M1))/M2 uses M1, the moment bound, not M2 or M3. Unless M2 ≤ 2^{-p} by accident, no T of that form makes the factor contract. The same disease appears in Appendix B, where T = 4τ + (4 log(2K1))/K2 gives 2K1 e^{-K2T} = e^{-4K2τ}/(8K1^3), not e^{-3K2T/4}. Both are load-bearing: this coefficient is exactly what makes the accumulated error over windows summable in (A.9)-(A.11). A repair is plain—choose T so that 2^p M2 e^{-M3 T} ≤ 1/2 and rerun the geometric series—but it is not in the paper.\n\nWhat is good: the abstraction into four preconditions (Markov-chain structure, infinite-horizon moment bound, exponential attraction, finite-time convergence) is a clean way to separate the window-patching argument from the method-specific finite-time analysis. The transfer to segment processes and, from strong convergence, to convergence of the numerical law toward the true invariant measure without separately proving a numerical invariant measure is a nice trick. The applications to TEM and BEM are credible: the finite-time lemmas in Sections 4 and 5 are standard but competently handled, and the literature is engaged (Crisan et al., Mao's group). The numerical experiments are illustrative only; no error bars, no code, but that is not the weakness here.\n\nThe paper is worth a serious referee if the T-selection defect is fixed. As it stands, Theorem 3.3 and Lemma 3.5 are unproved, and Theorem 3.6 inherits the failure. I would not cite the current version. For a reading group, the framework is useful to know about, but I'd present it with a caveat.\n\nRecommendation: send to peer review, but the referee should insist on correcting the window length before acceptance.","headline":"Useful abstraction, but the key contraction step in Appendices A and B is algebraically wrong; the proof as written does not establish the time-uniform bound.","tokens_in":36817,"tokens_out":3010,"would_cite":false,"duration_ms":28521,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-07T15:37:41.537132+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}