Pith. sign in

REVIEW 2 major objections 7 minor 28 references

Implicit numerical approximation for stochastic delay differential equations with the nonlinear diffusion term in the infinite horizon

T0 review · 2 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2505.12883 v1 pith:GWXUJI7E submitted 2025-05-19 math.NA cs.NAmath.PR

Yudong Wang , Hongjiong Tian This is my paper · ORCID
classification math.NAcs.NAmath.PR MSC 65C3060H1060H35
keywords stochasticdelaydifferentialequationsbackwardEuler–Maruyamainfinite-horizonconvergencestrongratenonlineardiffusioninvariantmeasuresegmentprocessKhasminskii-typeconditions
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

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.

What carries the argument

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 $2\log2/\lambda_2$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 7 minor

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.

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 (2)
  1. [§4, Theorem 4.4 proof, Eq. (4.9)] 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.
  2. [§4, Lemma 4.5 proof, Eqs. (4.13)–(4.17)] 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.
minor comments (7)
  1. [Assumption 2.1] 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).
  2. [Lemma 4.2] 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.
  3. [Lemma 4.2 vs. Assumption 2.3] 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.
  4. [Lemma 4.5 statement] 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.
  5. [Theorem 4.8 proof] 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.
  6. [Section 5] 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.
  7. [End of Theorem 4.4 proof] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the BEM infinite-horizon convergence and invariant-measure approximation are derived from explicit assumptions, with self-citations only as external prior results.

full rationale

The derivation chain is self-contained where circularity could hide. Theorem 4.4 is proved by combining Lemma 4.2 (finite-time strong convergence, proved from Assumptions 2.1-2.3 via moment estimates and Holder/Burkholder-Davis-Gundy bounds) with Lemma 4.3 (contractivity of BEM with respect to different initial segments, proved directly from Assumption 2.2). The block induction then gives a t-independent constant C, with no fitted parameter and no use of the target estimate as an input. Theorem 4.8 is a triangle-inequality argument: Lemma 4.7 converts the t-uniform pathwise closeness into bounded-Lipschitz distance, and Lemma 3.3 supplies the underlying invariant measure pi. The cited items that overlap with the authors, [25] and [27], are published external results used for standard facts (well-posedness/Markov property, invariant-measure existence for the exact segment process); they do not redefine the numerical error or assume the BEM convergence claim. In particular, Lemma 3.3 is not a re-statement of Theorem 4.4, and no quantity in the main theorems is fitted to data. I note, as a correctness (not circularity) concern, that the proof of Theorem 4.4 applies Lemma 4.2 to random initial segments x_T^{0,xi} in later blocks, although the lemma is stated for deterministic xi in B(R); this is a technical gap to be repaired, not a circular reduction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The main theorems rest on several imported results: well-posedness of the underlying SDDE, unique solvability of the implicit BEM step, and the existence of an invariant measure for the segment process. These are not proved here and are cited from prior work, including the authors' own [25] and [27].

free parameters (2)
  • lambda1 (decay rate in Lemma 4.1)
    Ad hoc constant chosen small enough to satisfy b10 - lambda1 - b11 e^{lambda1 tau} > 0 and b12 - b13 e^{lambda1 tau} > 0; any sufficiently small value works, so the results are not numerically fitted.
  • lambda2 (contraction rate in Lemma 4.3)
    Ad hoc constant chosen small enough to satisfy b1 - l1 a2 - (lambda2 or epsilon3) - (b2 + l1 a3) e^{lambda2 tau} > 0; existence is asserted, no numerical value is assigned.
assumptions (5)
  • domain assumption Under Assumptions 2.1 and 2.2, SDDE (2.1) has a unique global solution with E|x(t)|^{4q-2} finite for all t.
    Invoked to define the underlying solution and to bound moments in Lemmas 4.2 and Theorem 4.4 (cited from [19, Theorem 2.4] and [21]).
  • domain assumption The backward Euler-Maruyama scheme (2.3) is well defined, i.e., the implicit step for X_k has a unique solution for Delta > 0 under Assumption 2.2.
    The paper cites [25, Theorem 6.1] for this and does not prove it.
  • domain assumption The segment process {x_t} of the underlying solution is asymptotically stable in distribution and admits a unique invariant measure pi (Lemma 3.3).
    Stated without proof and cited from [27]; Theorem 4.8 uses it as the target measure.
  • domain assumption The numerical segment process {X_{t_k}} is a homogeneous Markov chain (Lemma 2.4).
    Cited from [25, Theorem 6.1]; needed to interpret invariant-measure convergence for the numerical scheme.
  • standard math Standard stochastic calculus inequalities (Young, Burkholder-Davis-Gundy, Gronwall) are used throughout.
    Routine tools; no explicit proof is needed.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Implicit numerical approximation for stochastic delay differential equations with the nonlinear diffusion term in the infinite horizon." pith.science (2026). https://pith.science/paper/GWXUJI7E

@misc{pith2026250512883,
  author       = {Pith},
  title        = {Pith review of: Implicit numerical approximation for stochastic delay differential equations with the nonlinear diffusion term in the infinite horizon},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GWXUJI7E}},
  note         = {Machine review of arXiv:2505.12883}
}
abstract

This paper investigates the approximation of stochastic delay differential equations (SDDEs) via the backward Euler-Maruyama (BEM) method under generalized monotonicity and Khasminskii-type conditions in the infinite horizon. First, by establishing the uniform moment boundedness and finite-time strong convergence of the BEM method, we prove that for sufficiently small step sizes, the numerical approximations strongly converge to the underlying solution in the infinite horizon with a rate of $1/2$, which coincides with the optimal finite-time strong convergence rate. Next, we establish the uniform boundedness and convergence in probability for the segment processes associated with the BEM method. This analysis further demonstrates that the probability measures of the numerical segment processes converge to the underlying invariant measure of the SDDEs. Finally, a numerical example and simulations are provided to illustrate the theoretical results.

Figures

Figures reproduced from arXiv: 2505.12883 by the authors.

Figure 1
Figure 1. The asymptotic behaviour of the error constant [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Sample means of E(|f(z(k))|) (k ∈ N) with different initial data ξ In this section, an example will be presented in order to verify the theories. Example 1. Consider the following SDDE  dx(t) = 10 − x(t) − 10x 3 (t)  dt + [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references · 5 canonical work pages

  1. [27]

    Y . Wang, F. Wu, and X. Mao. Stability in distribution of s tochastic functional differential equations. Systems Control Lett., 132, 2019

    2019

  2. [1]

    Angeli, D

    L. Angeli, D. Crisan, and M. Ottobre. Uniform in time conv ergence of numerical schemes for stochastic differen- tial equations via strong exponential stability: Euler met hods, split-step and tamed schemes. arXiv:2303.15463, 2023

    arXiv 2023

  3. [2]

    Arriojas, Y

    M. Arriojas, Y . Hu, S.-E. Mohammed, and G. Pap. A delayed b lack and scholes formula. Stochastic Anal. Appl., 25(2):471 – 492, 2007

    2007

  4. [3]

    J. Bao, J. Shao, and C. Y uan. Invariant probability measu res for path-dependent random diffusions. Nonlinear Anal., 228, 2023

    2023

  5. [4]

    E. Buckwar. Introduction to the numerical analysis of st ochastic delay differential equations. J. Comput. Appl. Math., 125(1-2):297 – 307, 2000

    2000

  6. [5]

    Cacace, A

    F. Cacace, A. Germani, C. Manes, and M. Papi. Predictor-b ased control of stochastic systems with nonlinear diffusions and input delay. Automatica, 107:43 – 51, 2019

    2019

  7. [6]

    Crisan, P

    D. Crisan, P . Dobson, and M. Ottobre. Uniform in time esti mates for the weak error of the Euler method for SDEs and a pathwise approach to derivative estimates for diffusion semigroups. Trans. Amer . Math. Soc., 374(5):3289– 3330, 2021

    2021

  8. [7]

    J. P . D´ ecamps, F. Gensbittel, and T. Mariotti. Investment timing and technological breakthroughs, 2021

    2021

Show all 28 references
  1. [8]

    Grenadier and A

    S. Grenadier and A. Weiss. Investment in technological i nnovations: An option pricing approach. Journal of Financial Economics, 44(3):397–416, 1997

    1997

  2. [9]

    Q. Guo, X. Mao, and R. Y ue. The truncated euler–maruyama m ethod for stochastic differential delay equations. Numer . Algorithms., 78(2):599 – 624, 2018

    2018

  3. [10]

    D. J. Higham, X. Mao, and A. M. Stuart. Exponential mean- square stability of numerical solutions to stochastic differential equations. LMS J. Comput. Math. , 6:297 – 313, 2003

    2003

  4. [11]

    Ikeda and S

    N. Ikeda and S. Watanabe. Stochastic differential equations and diffusion processes. North-Holland, Amsterdam, 1989

    1989

  5. [12]

    Ji and C

    Y . Ji and C. Y uan. Tamed em scheme of neutral stochastic d ifferential delay equations. J. Comput. Appl. Math. , 326:337 – 357, 2017

    2017

  6. [13]

    Lei and M

    J. Lei and M. C. Mackey. Stochastic differential delay e quation, moment stability, and application to hematopoi- etic stem cell regulation system. SIAM J. Appl. Math. , 67(2):387 – 407, 2006. 18 A PREPRINT - A UGUST 27, 2025

    2006

  7. [14]

    X. Li, X. Mao, and G. Song. Explicit approximation of inv ariant measure for stochastic delay differential equa- tions with the nonlinear diffusion term. J. Theoret. Probab., 2023

    2023

  8. [15]

    M. Liu, W . Cao, and Z. Fan. Convergence and stability of t he semi-implicit euler method for a linear stochastic differential delay equation. J. Comput. Appl. Math. , 171(1-2):255 – 268, 2004

    2004

  9. [16]

    X. Mao. Exponential stability of equidistant euler-ma ruyama approximations of stochastic differential delay equations. J. Comput. Appl. Math. , 200(1):297 – 316, 2007

    2007

  10. [17]

    X. Mao. Stochastic Differential Equations and Applications . Horwood, Chichester, UK, 2 edition, 2007

    2007

  11. [18]

    X. Mao. Almost sure exponential stability in the numeri cal simulation of stochastic different equations. SIAM J. Numer . Anal., 53(1):370 – 389, 2015

    2015

  12. [19]

    Mao and M

    X. Mao and M. J. Rassias. Khasminskii-type theorems for stochastic differential delay equations. Stochastic Anal. Appl., 23(5):1045 – 1069, 2005

    2005

  13. [20]

    Mao and S

    X. Mao and S. Sabanis. Numerical solutions of stochasti c differential delay equations under local lipschitz condition. J. Comput. Appl. Math. , 151(1):215 – 227, 2003

    2003

  14. [21]

    Mao and C

    X. Mao and C. Y uan. Stochastic Differential Equations with Markovian Switchi ng. Imperial College Press, London, 2 edition, 2006

    2006

  15. [22]

    E. J. McShane. Stochastic Calculus and Stochastic Models. Academic . Academic Press, 1974

    1974

  16. [23]

    Mohammed

    S.E.A. Mohammed. Stochastic Functional Differential Equations. Longman, New Y ork, 1986

    1986

  17. [24]

    M. Reiß, M. Riedle, and O. van Gaans. Delay differential equations driven by l´ evy processes: stationarity and feller properties. Stochastic Process. Appl., 116(10):1409 –1432, 2006

    2006

  18. [25]

    B. Shi, Y . Wang, X. Mao, and F. Wu. Approximation of invar iant measures of a class of backward euler- maruyama scheme for stochastic functional differential eq uations. J. Differential Equations , 389:415 – 456, 2024

    2024

  19. [26]

    G. Song, J. Hu, S. Gao, and X. Li. The strong convergence a nd stability of explicit approximations for nonlinear stochastic delay differential equations. Numer . Algorithms, 89(2):855 – 883, 2022

    2022

  20. [28]

    S. Zhou. Strong convergence and stability of backward e uler–maruyama scheme for highly nonlinear hybrid stochastic differential delay equation. Calcolo, 52(4):445 – 473, 2015. 19

    2015

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.