{"id":"8f782c2b-3aa1-4f9a-a7cf-d31c94339527","arxiv_id":"2505.12883","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For stochastic delay differential equations under monotonicity, the backward Euler-Maruyama method converges strongly at rate 1/2 uniformly in time and its segment processes converge in law to the true invariant measure.","lead":"This paper proves that the backward Euler-Maruyama numerical method keeps a constant error bound when simulating stochastic delay equations over arbitrarily long time horizons. It gives a theoretical guarantee that long-run simulations of noisy systems with memory can use a fixed step size, and that the simulated statistics converge to the true steady state.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.4 applies finite-time Lemma 4.2 to random initial segments x_{jT}^{0,ξ} at every block after the first, though Lemma 4.2 is stated and proved only for deterministic ξ in B(R); without a random-initial-data version the block induction does not close.","rationale":"The paper's central claim is the uniform-in-time strong convergence bound of Theorem 4.4. That theorem is proved by partitioning the half-line into fixed-length blocks and at each block restarting the BEM from the exact solution segment at the block's left endpoint. The restart makes the argument formally depend on Lemma 4.2 with random initial data, but Lemma 4.2 is a deterministic-initial statement with ξ in B(R), and its proof is written for that case. The reader's own rationale names this as an unstated reliance on random-initial-data versions of Lemma 4.2, even though the formal weakest_assumption field selects the quantitative dissipativity condition. I agree with the reader's conditional assessment: the gap is real and load-bearing, but it is of the kind that standard moment and conditional arguments can close, so I would not reject outright. The alternative concerns are weaker: the quantitative dominance needed for λ2 is automatically implied by Assumption 2.2; Lemma 3.3 is imported from a published source; and the double limit in Theorem 4.8 can be made uniform because the construction in Lemma 4.7 does not actually depend on k. Thus the reader's CONDITIONAL verdict stands without adjustment.","tokens_in":23514,"tokens_out":19615,"duration_ms":196935,"concrete_test":"Formally state and prove Lemma 4.2R: for F0-measurable random initial segments ξ with E||ξ||^{4q-2} < ∞, E|x(t_k; 0, ξ) - X_k^{0,ξ}|^2 ≤ C_T (1 + sup_{t ≥ -τ} E|x(t; 0, ξ)|^{4q-2}) Δ for 0 ≤ k ≤ T/Δ, with C_T and the Δ-threshold independent of ω. Replace Assumption 2.3 by the t ≥ 0 modulus estimate used in the proof of Lemma 4.2, or by Lemma 3.2 if initial regularity is needed. Then audit every application of Lemma 4.2 in the block induction of Theorem 4.4, including display (4.9), to check that it is covered by Lemma 4.2R and that the threshold does not depend on the block index j. If the random-initial version fails, the uniform O(Δ) bound is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 4.4 is an induction over blocks [jT,(j+1)T] with T = 2 log 2 / λ2 + 1. For j = 0, Lemma 4.2 is applied with the original deterministic segment ξ in B(R). For every later block the same lemma is invoked with initial segment x_{jT}^{0,ξ}, i.e. with x(·; 0, ξ) restricted to [jT - τ, jT]. That segment is F_{jT}-measurable, is not contained in B(R), and its modulus of continuity is controlled only probabilistically by Lemma 3.2, not by Assumption 2.3 as used in Lemma 4.2. The paper neither states a random-initial-data formulation of Lemma 4.2 nor provides a conditional or stopping argument that would allow replacing the deterministic bounded initial segment by the random exact segment. Display (4.9) and the block recursion that follows depend on exactly this replacement. Consequently the recursive estimate E|x(t_k; 0, ξ) - X_k^{0,ξ}|^2 ≤ CΔ does not close over the infinitely many blocks, so Theorem 4.4 is not established as written. Lemma 4.7 and Theorem 4.8 inherit this gap because they use Theorem 4.4. The quantitative dissipativity condition highlighted by the reader is not the weakest point: Assumption 2.2 already gives b1 > b2 + l1(a2 + a3) and b3 > l1(a4 + a5), so by continuity a small λ2 satisfying the inequalities in Lemmas 4.2 and 4.3 exists. The missing random-initial-data lemma is likely provable from the moment bounds already derived, but it is load-bearing and must be supplied.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the backward Euler–Maruyama (BEM) scheme for stochastic delay differential equations with nonlinear drift and diffusion, under generalized monotonicity and Khasminskii-type conditions. The main claims are: (i) uniform moment boundedness of the numerical solution (Lemma 4.1); (ii) finite-time strong convergence of order 1/2 (Lemma 4.2); (iii) strong convergence uniformly in time on the infinite horizon for sufficiently small step size (Theorem 4.4); (iv) uniform boundedness in probability and convergence in probability of the numerical segment process (Lemmas 4.5–4.7); and (v) convergence of the probability measures of the numerical segment process to the underlying invariant measure in the bounded-Lipschitz metric (Theorem 4.8). A numerical example for a scalar cubic-drift SDDE is provided.","tokens_in":23879,"tokens_out":7543,"duration_ms":77210,"significance":"If established, Theorem 4.4 and Theorem 4.8 would extend uniform-in-time strong convergence and invariant-measure approximation results from SDEs to SDDEs with non-globally Lipschitz coefficients, which is a useful direction given the prior work of Crisan et al. and recent SDDE invariant-measure papers. The paper's strategy of exploiting the contractivity of the implicit scheme through Lemma 4.3 is sensible, and the goal of an error constant independent of t is well motivated. The proofs import several key ingredients from prior work (Lemma 3.3 from [27], well-posedness and Markov properties from [19,25]), which is acceptable, but it means the novel part rests on the block-induction argument in Theorem 4.4 and on the segment-process lemmas. Both currently contain gaps, so the contribution is conditional.","major_comments":[{"comment":"The block-induction argument applies Lemma 4.2, which is stated and proved for deterministic ξ in B(R), to the random initial segments x_{jT}^{0,ξ} for j ≥ 1. Those segments are F_{jT}-measurable, are not contained in B(R), and their modulus of continuity is controlled only probabilistically by Lemma 3.2, not by Assumption 2.3 as required in Lemma 4.2. Equation (4.9) and the subsequent block recursion depend on exactly this replacement, so the recursive estimate E|x(t_k;0,ξ) - X_k^{0,ξ}|^2 ≤ C Δ does not close over infinitely many blocks as written. A random-initial-data version of Lemma 4.2, or a conditional or stopping argument with constants independent of the random segment, must be supplied without it, Theorem 4.4 is not established, and Lemma 4.7 and Theorem 4.8 inherit the gap.","section":"§4, Theorem 4.4 proof, Eq. (4.9)"},{"comment":"The proof of Lemma 4.5 does not establish the claimed uniform bound for the numerical segment process. The stopping-time estimate (4.13) controls only discrete-time values up to (k+N) ∧ λ, whereas the lemma requires control of the interpolated process on the entire interval [t_k, t_k+T], and no bound on the increments of the interpolated process is used. In addition, the implication preceding (4.15) has the wrong direction, and the displayed inequality after (4.17) is reversed: the event {‖X_t^{0,ξ}‖ ≤ H} is contained in, not containing, {|X(t;0,ξ)| ≤ H}. The appeal to Lemma 3.1 there concerns the underlying solution, not the numerical solution. Since Lemma 4.5 is used in Lemma 4.6 and Lemma 4.7, this gap is load-bearing for the segment-process convergence result.","section":"§4, Lemma 4.5 proof, Eqs. (4.13)–(4.17)"}],"minor_comments":[{"comment":"Several displays in Assumption 2.1 have missing overbars and arguments: for example, |g(x,y) - g(x,y)|^2 and |g_1(x) - g_1(x)|^2 should be differences with barred arguments, and V(x,x) in Assumption 2.2 should be V(x, x̄) (or the bar notation introduced consistently).","section":"Assumption 2.1"},{"comment":"The statement of Lemma 4.2 says 'where ǫ3 is a positive constant', but the proof uses ǫ6; Lemma 4.3 also uses an ǫ3 that is not defined in that lemma. The notation should be harmonized.","section":"Lemma 4.2"},{"comment":"Lemma 4.2 is stated under Assumptions 2.1 and 2.2 only, but its proof explicitly invokes Assumption 2.3 after Eq. (4.5). If Assumption 2.3 is needed for the finite-time result, it should be stated in Lemma 4.2; otherwise the proof should explain why it is not needed.","section":"Lemma 4.2 vs. Assumption 2.3"},{"comment":"The statement uses the notation ‖X_{t_k}^{0,ξ}‖ with the quantifier 'for all s ∈ [t_k, t_k+T]'; the norm should apply to X_s^{0,ξ}, not to a single segment indexed by t_k.","section":"Lemma 4.5 statement"},{"comment":"The proof writes d_L(P_{t_k}(ξ,·), P_{t_k}(ξ,·)) for the distance between the exact and numerical segment distributions, using the same symbol for both measures; different notation for the exact and numerical transition probabilities would avoid confusion.","section":"Theorem 4.8 proof"},{"comment":"The numerical experiment treats the fine-step numerical solution with Δ_1 = 0.0001 as the exact solution and gives no confidence intervals or path counts for the K-S statistics; this illustrates the qualitative behavior but is not a quantitative verification of the theorems.","section":"Section 5"},{"comment":"The sentence 'By the arbitrariness of j and Lemma 1' should refer to a numbered lemma in the paper; as written, 'Lemma 1' is undefined.","section":"End of Theorem 4.4 proof"}],"recommendation":"major_revision","confidential_remarks":"The gap concerning random initial data in Theorem 4.4 is likely repairable: a random-initial-data version of Lemma 4.2 should follow from the same estimates once the constants are expressed in terms of sup_t E|x(t;0,η)|^{4q-2} for η in a bounded set, together with the Markov property. I therefore recommend major revision rather than rejection. The paper's heavy reliance on Lemmas from [25] and [27] should also be checked for overlap and proper attribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this paper extends finite-time BEM strong convergence to infinite horizon for SDDEs with nonlinear diffusion, using a block argument plus segment-process tightness, and proves the numerical segment measures converge to the invariant measure. The main theorems are new relative to the literature, and the strategy is recognizable: coupling contraction from the dissipativity assumptions plus finite-time error estimates. If the proof closes, it removes the exponential-in-T error blow-up for a practically relevant class of delay equations.\n\nWhat it does well: the moment bounds in Lemma 4.1 are carefully done; the coupling Lemma 4.3 is the right tool; the use of the segment process to bridge to invariant measure convergence is sensible. Credit where due: the paper does not fake the numerics, and the theoretical claims are clearly stated.\n\nThe soft spot is real. Theorem 4.4's proof is an induction over blocks of length T, and for every block after the first it invokes Lemma 4.2 with the random segment x_{jT}^{0,xi} as initial data. Lemma 4.2 is stated and proved only for deterministic xi in B(R), using Assumption 2.3 for the Holder constant. The segment x_{jT}^{0,xi} is F_{jT}-measurable, not in B(R), and its modulus of continuity is only controlled probabilistically by Lemma 3.2. The paper does not provide a random-initial-data version of Lemma 4.2, nor a conditioning or stopping argument. So the estimate in (4.9) does not close as written, and Lemma 4.7 and Theorem 4.8 inherit the gap. I read the gap as fixable rather than fatal: the moment bounds in Lemma 4.1 and Lemma 3.2 are probably enough to prove a conditioned version, but it is load-bearing and must be written out.\n\nThe reader's worry about the quantitative dissipativity condition is less serious: Assumption 2.2 already gives strict inequalities, so a small lambda2 exists by continuity. The typos (epsilon3/epsilon6 confusion, misplaced overbars, Lemma 4.5 statement) are annoying but not substantive.\n\nBottom line: the paper deserves a serious referee. I would send it out with a request to fix the random-initial-data gap and clean up the notation, and to state precisely which lemmas are imported and in what form. No code or data accompanies the numerical claims, which is fine for an illustration but means the experimental part adds little evidential weight.\n\nRecommendation: engage with it, but condition acceptance on a corrected block induction.","headline":"A plausible infinite-horizon strong convergence result for BEM on SDDEs, with a genuine but likely fixable gap in the block induction: Lemma 4.2 is applied to random initial segments.","tokens_in":24436,"tokens_out":2479,"would_cite":false,"duration_ms":25390,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65C30","60H10","60H35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that backward Euler–Maruyama retains the optimal 1/2 strong convergence rate uniformly for all time for stochastic delay differential equations with nonlinear diffusion, and that its numerical segment processes converge…","keywords":["stochastic delay differential equations","backward Euler–Maruyama","infinite-horizon convergence","strong convergence rate","nonlinear diffusion","invariant measure","segment process","Khasminskii-type conditions"],"falsifier":"For the linear test SDDE $dx(t)=(-b x(t)+c x(t-\\tau))dt+\\sigma x(t-\\tau)dW(t)$, run BEM with a fixed step $\\Delta$ and initial segment $x(\\theta)=\\cos\\theta$, measuring $\\mathbb{E}|x(t)-X(t)|^2$ by Monte Carlo or by solving the exact covariance recursion at horizons $t=10,100,1000$. If the ratio of this mean-square error to $\\Delta$ grows with $t$ for coefficient choices that satisfy all stated inequalities, the uniform bound of Theorem 4.4 fails. A complementary check is whether the constants in Assumption 2.2 can be verified from the coefficients at all; the paper offers no algorithm or worked verification for this, so any model where the margin cannot be certified is outside the theorem.","tokens_in":23261,"feed_emoji":"📈","tokens_out":8605,"duration_ms":83237,"temperature":0.7,"pith_summary":"The paper proves that the backward Euler–Maruyama (BEM) method—the standard implicit Euler scheme for stochastic differential equations—keeps its strong convergence rate of 1/2 uniformly in time when applied to scalar stochastic delay differential equations with nonlinear drift and diffusion coefficients. Under generalized monotonicity and Khasminskii-type conditions, the mean-square error between the true solution and the BEM approximation at every time grid point is bounded by $C\\Delta$ with a constant $C$ that does not grow with time. It then shows that the path-segment processes generated by BEM converge in probability to the true solution's segments, and that their probability measures converge to the solution's invariant measure in the bounded-Lipschitz metric. A sympathetic reader would care because this removes the usual requirement to shrink the step size as the simulation horizon grows, so long-time Monte Carlo studies of randomly perturbed delayed systems can trust one fixed step size.","feed_headline":"Backward Euler keeps its 1/2 error rate forever","feed_subtitle":"The implicit scheme keeps order-1/2 mean-square error at all times, and its segment laws reach the invariant measure.","key_machinery":"The load-bearing object is the backward Euler–Maruyama map together with the exponential contraction inequality of Lemma 4.3. The map defines each new state implicitly by $X_k=X_{k-1}+f(X_k,X_{k-M})\\Delta+g(X_{k-1},X_{k-M-1})\\Delta W_{k-1}$; this implicit structure is what lets the scheme inherit the dissipativity of the drift. The contraction inequality says that two BEM paths started from different initial segments satisfy a weighted mean-square difference bounded by the initial difference times $(1+\\lambda_2\\Delta)^{-k}$ plus negative accumulative terms, where $\\lambda_2$ must satisfy $b_1-l_1a_2-(\\lambda_2\\vee\\epsilon_3)-(b_2+l_1a_3)e^{\\lambda_2\\tau}>0$ and $b_3-l_1a_4-l_1a_5 e^{\\lambda_2\\tau}>0$. This geometric forgetting of the initial data is what converts a finite-time error estimate into a uniform-in-time estimate, by stitching over blocks of length roughly $2\\log2/\\lambda_2$.","core_discovery":"For the scalar SDDE $dx(t)=f(x(t),x(t-\\tau))dt+g(x(t),x(t-\\tau))dW(t)$ with initial segment $\\xi$, the paper establishes two statements. Theorem 4.4: for all sufficiently small step sizes $\\Delta$, the BEM approximation satisfies $\\mathbb{E}|x(t_k;0,\\xi)-X_k^{0,\\xi}|^2\\le C\\Delta$ for every grid point $t_k$, with $C$ independent of $t_k$ though dependent on the initial segment; this is the optimal finite-time strong rate of 1/2, now holding uniformly over the infinite horizon. Theorem 4.8: the probability measures of the BEM segment processes converge to the unique invariant measure $\\pi$ of the underlying SDDE in the bounded-Lipschitz metric as $k\\to\\infty$ and $\\Delta\\to0$. The route is: uniform moment bounds for BEM, a finite-time strong error estimate for BEM, a contraction inequality saying two BEM paths started from different initial segments merge exponentially fast, and a patching argument that propagates the finite-time error bound to arbitrarily large times.","pith_inferences":["If the paper is right, a practical corollary is that one fixed backward Euler step size can serve for both burn-in and sampling in long-run simulations, because the error bound is uniform across early and late times.","Because the proof needs only a one-step contraction plus a finite-time strong error estimate, the same block-patching argument should extend to split-step, tamed, or truncated explicit schemes whenever they admit an analogous contraction; this is an extension proposal, not a claim in the paper.","The proof is developed in the scalar setting; whether it survives vector-valued SDDEs or non-equidistant delays is open, since the segment-process and exponential-weighting estimates would need reworking.","A user who cannot compute the constants in Assumption 2.2 faces a hypothesis that is hard to verify in practice; a numerical procedure for estimating the dissipativity margin from short pilot runs would be a natural companion tool."],"forward_implications":["For a fixed step size $\\Delta$, the BEM error remains of order $\\Delta$ at arbitrarily late times, so long-horizon simulations do not need to shrink the step size as the terminal time grows.","The numerical segment processes converge in probability to the true solution's segments, so path-dependent statistics over the delay window are inherited by the simulation.","The law of the BEM segment process converges to the underlying invariant measure in the bounded-Lipschitz metric, meaning empirical averages over one long BEM path can approximate expectations under the invariant measure.","The geometric contraction of the numerical paths implies the numerical solution is asymptotically stable in distribution, matching the ergodic behavior of the underlying SDDE."],"supporting_citations":[{"why":"Supplies the Khasminskii-type existence theorem guaranteeing a unique global solution with finite high-order moments, which the convergence proof assumes.","marker":"[19]"},{"why":"Provides the textbook existence and moment-boundedness result for this class of stochastic functional differential equations, used alongside [19].","marker":"[21]"},{"why":"Establishes that the solution segment process is a homogeneous Markov process, the starting point for the invariant-measure analysis.","marker":"[23]"},{"why":"Provides the stability-in-distribution framework used to conclude that the underlying segment process admits a unique invariant measure.","marker":"[27]"},{"why":"Shows that the backward Euler–Maruyama scheme generates a homogeneous Markov chain, making the numerical segment process amenable to the same analysis.","marker":"[25]"},{"why":"Earlier approximation of invariant measures for SDDEs with nonlinear diffusion by an explicit scheme; the paper's segment-process convergence argument extends this to the implicit scheme.","marker":"[14]"},{"why":"Uniform-in-time weak error analysis for Euler-type schemes; cited as the approach the paper adapts to obtain strong convergence rather than weak convergence.","marker":"[1]"},{"why":"Establishes uniform-in-time weak convergence methods for Euler schemes, motivating the infinite-horizon setting the paper treats in a strong sense.","marker":"[6]"},{"why":"Establishes finite-time strong convergence and stability of backward Euler–Maruyama for highly nonlinear hybrid SDDEs, the local estimate that Theorem 4.4 propagates to infinite time.","marker":"[28]"}],"fun_headline_variants":["BEM maintains 1/2 strong error for all time","Backward Euler: uniform 1/2 error on infinite horizon","BEM segment laws converge to invariant measure","Order-1/2 strong error holds over all time in SDDEs","BEM's 1/2 rate persists across the infinite horizon"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The quantitative dissipativity condition in Assumption 2.2—that the drift's one-sided Lipschitz constant beats the delay feedback and nonlinear diffusion constants by a margin that also survives multiplication by $e^{\\lambda_2\\tau}$—must actually hold, and the paper does not explain how a user would verify the constants for a given model.","fun_headline_variants_meta":{"raw":{"variants":["BEM maintains 1/2 strong error for all time","Backward Euler: uniform 1/2 error on infinite horizon","BEM segment laws converge to invariant measure","Order-1/2 strong error holds over all time in SDDEs","BEM's 1/2 rate persists across the infinite horizon"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001113,"raw_usage":{"total_tokens":4628,"prompt_tokens":932,"completion_tokens":3696,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":548,"completion_tokens_details":{"reasoning_tokens":3609}},"tokens_in":548,"tokens_out":3696,"duration_ms":25073,"temperature":1.0,"reasoning_tokens":3609,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:25:23.415308+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the linear test SDDE $dx(t)=(-b x(t)+c x(t-\\tau))dt+\\sigma x(t-\\tau)dW(t)$, run BEM with a fixed step $\\Delta$ and initial segment $x(\\theta)=\\cos\\theta$, measuring $\\mathbb{E}|x(t)-X(t)|^2$ by Monte Carlo or by solving the exact covariance recursion at horizons $t=10,100,1000$. If the ratio of this mean-square error to $\\Delta$ grows with $t$ for coefficient choices that satisfy all stated inequalities, the uniform bound of Theorem 4.4 fails. A complementary check is whether the constants in Assumption 2.2 can be verified from the coefficients at all; the paper offers no algorithm or worked verification for this, so any model where the margin cannot be certified is outside the theorem.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that the backward Euler–Maruyama scheme generates a homogeneous Markov chain, making the numerical segment process amenable to the same analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes finite-time strong convergence and stability of backward Euler–Maruyama for highly nonlinear hybrid SDDEs, the local estimate that Theorem 4.4 propagates to infinite time."}],"review_version":1}