Pith. sign in

REVIEW 1 major objections 2 minor 21 references

Weak Error Estimates of Ergodic Approximations for Monotone Jump-diffusion SODEs

T0 review · 1 major / 2 minor · reviewed 2026-05-18 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read Backward Euler method achieves order-one convergence to invariant measures for monotone SODEs without jumps.

desk verdict They deliver a time-independent O(h) weak error between exact and numerical invariant measures for the backward Euler method on monotone jump-free SODEs, directly answering their own prior open question. read the letter →

arxiv 2509.15698 v2 submitted 2025-09-19 math.NA cs.NAmath.PR

Zhihui Liu , Xiaoming Wu This is my paper · ORCID
classification math.NAcs.NAmath.PR
keywords weakerrorestimatesexponentialergodicityinvariantmeasuresbackwardEulermethodstochasticthetajump-diffusionSODEsmonotonecoefficientsdissipativecondition
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

This paper establishes exponential ergodicity for the stochastic theta method applied to monotone jump-diffusion stochastic ordinary differential equations under a dissipative condition on the coefficients. It then provides weak error estimates specifically for the backward Euler method, which corresponds to the theta method with parameter equal to one. The time-independent nature of these estimates in the case without jumps leads to a first-order rate of convergence between the exact invariant measure and the one generated by the numerical scheme. This result answers a question posed in previous research on ergodic approximations for such equations.

What carries the argument

Dissipative condition on the coefficients combined with the implicit theta-method discretization for θ=1, which controls the error in the long-time limit.

What would settle it

If the weak error between the exact and numerical solutions grows with time instead of staying bounded, or if the convergence rate of invariant measures is observed to be lower than one in numerical tests for jump-free cases.

Watch

Extended reading notes

Core claim

For monotone jump-diffusion SODEs satisfying a dissipative condition, the stochastic theta method with θ ∈ (1/2,1] is exponentially ergodic. The backward Euler method yields weak error estimates, and in the jump-free case a time-independent bound gives one-order convergence of the numerical invariant measure to the exact one.

Load-bearing premise

The coefficients of the SODE satisfy a dissipative condition that ensures monotonicity.

Editorial extensions

If this is right

  • The stochastic theta method with θ > 1/2 is exponentially ergodic for the given class of equations.
  • Weak error estimates hold for the backward Euler method applied to these SODEs.
  • In the absence of jumps, the invariant measures converge at order one as the time step decreases.
  • Error bounds remain independent of simulation time for the backward Euler method in the jump-free case.

Reading between the lines

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

  • These time-independent bounds could support more reliable long-time statistics in simulations of systems where jumps are absent.
  • The approach might extend to checking ergodicity preservation in other implicit discretizations under similar monotonicity assumptions.
Share X Bluesky LinkedIn Reddit HN

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

1 major / 2 minor

Summary. The manuscript derives exponential ergodicity for the stochastic theta method (STM, θ ∈ (1/2,1]) applied to monotone jump-diffusion SODEs under a dissipative condition on the coefficients. It then obtains weak error estimates for the backward Euler method (BEM, θ=1), with the key result being a time-independent weak error bound in the jump-free case that implies a first-order convergence rate between the exact invariant measure μ and the numerical invariant measure μ_h, thereby resolving an open question posed in Liu and Liu (J. Sci. Comput. 2025).

Significance. If the central derivations are correct, the work supplies rigorous, time-uniform weak error control for ergodic approximations of monotone SODEs, which is a useful advance for numerical analysis of long-time behavior in jump-diffusions. The explicit resolution of the prior open question on invariant-measure convergence rates is a clear strength, as is the separation of the ergodicity analysis (for general θ) from the error analysis (specialized to BEM).

major comments (1)
  1. [Weak error estimates for BEM (around the statement of the time-independent bound)] The passage from the finite-time weak error bound |E[φ(X_t)] − E[φ(Y_t)]| ≤ C h to the time-independent bound |μ(φ) − μ_h(φ)| ≤ C h (jump-free case) requires that the prefactor C remain independent of t. The dissipative/monotonicity condition is used to obtain exponential ergodicity of both the exact process and the BEM separately, but it is not immediately clear that this yields a uniform-in-time Lyapunov control or Gronwall estimate on the difference process that prevents prefactor growth before the contraction takes over. A concrete verification of this uniform control (e.g., via a t-independent moment bound on the error or an explicit comparison of the one-sided Lipschitz and dissipativity constants) is needed to support the headline claim.
minor comments (2)
  1. Clarify the precise statement of the dissipative condition (one-sided Lipschitz plus growth) and confirm it is identical for both the ergodicity and error sections.
  2. Add a short remark on how the jump terms are handled in the error analysis versus the pure diffusion case, to make the distinction between the two settings explicit.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comment. We address it point by point below.

read point-by-point responses

  1. Referee: [Weak error estimates for BEM (around the statement of the time-independent bound)] The passage from the finite-time weak error bound |E[φ(X_t)] − E[φ(Y_t)]| ≤ C h to the time-independent bound |μ(φ) − μ_h(φ)| ≤ C h (jump-free case) requires that the prefactor C remain independent of t. The dissipative/monotonicity condition is used to obtain exponential ergodicity of both the exact process and the BEM separately, but it is not immediately clear that this yields a uniform-in-time Lyapunov control or Gronwall estimate on the difference process that prevents prefactor growth before the contraction takes over. A concrete verification of this uniform control (e.g., via a t-independent moment bound on the error or an explicit comparison of the one-sided Lipschitz and dissipativity constants) is needed to support the headline claim.

    Authors: We appreciate this observation, which highlights a point that merits explicit clarification. In the proof of the finite-time weak error bound (Theorem 4.2), the monotonicity assumption (Assumption 2.1) is used to derive a differential inequality for the error process that includes a dissipative drift term with constant α > 0. Because the one-sided Lipschitz constant satisfies a relation compatible with α (specifically, the effective contraction rate remains negative), the resulting Gronwall estimate yields a prefactor that is bounded independently of t; the bound does not grow before the exponential contraction dominates. We will add a short auxiliary lemma (new Lemma 4.3) that records the t-uniform moment bound on the error and makes the comparison of constants explicit. This lemma will directly justify the passage to the time-independent estimate in Corollary 4.4. The revision will therefore strengthen the presentation without altering the stated results. revision: yes

Circularity Check

1 steps flagged · score 2.0 of 10

Minor self-citation to prior work whose open question is resolved by independent estimates.

  1. self citation load bearing [Abstract]
    "answering a question left in Z. Liu and Z. Liu, J. Sci. Comput. (2025) 103:87"

    The citation frames the contribution as resolving a prior open question from overlapping-author work, but the citation is contextual only; the time-independent weak error estimate and one-order convergence for invariant measures are derived anew in this manuscript and do not reduce to any equation or constant from the cited paper.

full rationale

The manuscript derives exponential ergodicity for the stochastic theta method and weak error bounds for the backward Euler method directly from the dissipative/monotonicity condition on the SODE coefficients. These derivations are presented as self-contained within the current paper. The sole self-citation appears only to identify the open question being answered and does not supply any load-bearing step, fitted parameter, or uniqueness result that the new estimates reduce to by construction. No equation in the provided abstract or context equates a new prediction to a prior fitted quantity or renames an input as output.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The results rest on standard dissipativity/monotonicity assumptions for SODE coefficients and on the usual Lipschitz or growth conditions needed for existence of invariant measures; no new free parameters or invented entities are introduced.

assumptions (1)
  • domain assumption Dissipative condition on the drift and diffusion coefficients that guarantees exponential ergodicity of both the continuous and discrete processes.
    Invoked to obtain both the ergodicity of the stochastic theta method and the time-independent error bound.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Weak Error Estimates of Ergodic Approximations for Monotone Jump-diffusion SODEs." pith.science (2026). https://pith.science/paper/2509.15698

@misc{pith2026250915698,
  author       = {Pith},
  title        = {Pith review of: Weak Error Estimates of Ergodic Approximations for Monotone Jump-diffusion SODEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2509.15698}},
  note         = {Machine review of arXiv:2509.15698}
}
abstract

We first derive the exponential ergodicity of the stochastic theta method (STM) with $\theta \in (1/2,1]$ for monotone jump-diffusion stochastic ordinary differential equations (SODEs) under a dissipative condition. Then we establish the weak error estimates of the backward Euler method (BEM), corresponding to the STM with $\theta=1$. In particular, the time-independent estimate for the BEM in the jump-free case yields a one-order convergence rate between the exact and numerical invariant measures, answering a question left in {\it Z. Liu and Z. Liu, J. Sci. Comput. (2025) 103:87}.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    Cerrai,Second order PDE’s in finite and infinite dimension, Lecture Notes in Mathematics, vol

    S. Cerrai,Second order PDE’s in finite and infinite dimension, Lecture Notes in Mathematics, vol. 1762, Springer-Verlag, Berlin, 2001, A probabilistic approach. MR 1840644

    work page 2001

  2. [2]

    Chen and S

    Z. Chen and S. Gan,Convergence and stability of the backward Euler method for jump- diffusion SDEs with super-linearly growing diffusion and jump coefficients, J. Comput. Appl. Math.363(2020), 350–369. MR 3974743

    work page 2020

  3. [3]

    Z. Chen, S. Gan, and X. Wang,Mean-square approximations of L´ evy noise driven SDEs with super-linearly growing diffusion and jump coefficients, Discrete Contin. Dyn. Syst. Ser. B24 (2019), no. 8, 4513–4545. MR 3986287

    work page 2019

  4. [4]

    Cont and P

    R. Cont and P. Tankov,Financial modelling with jump processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004. MR 2042661

    work page 2004

  5. [5]

    Da Prato and J

    G. Da Prato and J. Zabczyk,Ergodicity for infinite-dimensional systems, London Mathe- matical Society Lecture Note Series, vol. 229, Cambridge University Press, Cambridge, 1996. MR 1417491

    work page 1996

  6. [6]

    S. Deng, W. Fei, W. Liu, and X. Mao,The truncated EM method for stochastic differential equations with Poisson jumps, J. Comput. Appl. Math.355(2019), 232–257. MR 3915385

    work page 2019

  7. [7]

    Y. Guo, X. Wang, and Y. Wu,Order-one convergence of the backward Euler method for random periodic solutions of semilinear SDEs, Discrete Contin. Dyn. Syst. Ser. B30(2025), no. 9, 3222–3242. MR 4893096

    work page 2025

  8. [8]

    Gy¨ ongy and N

    I. Gy¨ ongy and N. V. Krylov,On stochastic equations with respect to semimartingales. I, Stochastics4(1980/81), no. 1, 1–21. MR 587426

    work page 1980

Show all 21 references
  1. [9]

    D. J. Higham and P. E. Kloeden,Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems, J. Comput. Appl. Math.205(2007), no. 2, 949–956. MR 2329668

    2007

  2. [10]

    M. Li, C. Huang, and Z. Chen,Compensated projected Euler-Maruyama method for stochastic differential equations with superlinear jumps, Appl. Math. Comput.393(2021), Paper No. 125760, 11. MR 4176960

    2021

  3. [11]

    W. Liu, X. Mao, and Y. Wu,The backward Euler-Maruyama method for invariant measures of stochastic differential equations with super-linear coefficients, Appl. Numer. Math.184 (2023), 137–150. MR 4499301

    2023

  4. [12]

    Liu,Numerical ergodicity of stochastic Allen-Cahn equation driven by multiplicative white noise, Commun

    Z. Liu,Numerical ergodicity of stochastic Allen-Cahn equation driven by multiplicative white noise, Commun. Math. Res.41(2025), no. 1, 30–44. MR 4896563

    2025

  5. [13]

    Liu and Z

    Z. Liu and Z. Liu,Numerical unique ergodicity of monotone SDEs driven by nondegenerate multiplicative noise, J. Sci. Comput.103(2025), no. 3, Paper No. 87, 21. MR 4899897

    2025

  6. [14]

    Liu and X

    Z. Liu and X. Wu,Geometric ergodicity and strong error estimates for tamed schemes of super-linear SODEs, arXiv:2411.06049 (2024)

    2024

  7. [15]

    J. C. Mattingly, A. M. Stuart, and D. J. Higham,Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stochastic Process. Appl.101(2002), no. 2, 185–232. MR 1931266

    2002

  8. [16]

    Mikuleviˇ cius and C

    R. Mikuleviˇ cius and C. Zhang,On the rate of convergence of weak Euler approximation for nondegenerate SDEs driven by L´ evy processes, Stochastic Process. Appl.121(2011), no. 8, 1720–1748. MR 2811021

    2011

  9. [17]

    Pag` es and A

    G. Pag` es and A. Sagna,Weak and strong error analysis of recursive quantization: a general approach with an application to jump diffusions, IMA J. Numer. Anal.41(2021), no. 4, 2668–2707. MR 4328397

    2021

  10. [18]

    C. Pang, X. Wang, and Y. Wu,Linear implicit approximations of invariant measures of semi-linear SDEs with non-globally Lipschitz coefficients, J. Complexity83(2024), Paper No. 101842, 45. MR 4721566

    2024

  11. [19]

    S. Rong,Theory of stochastic differential equations with jumps and applications, Mathemat- ical and Analytical Techniques with Applications to Engineering, Springer, New York, 2005, Mathematical and analytical techniques with applications to engineering. MR 2160585

    2005

  12. [20]

    Y. Wang, Z. Chen, M. Niu, and Y. Niu,Mean-square convergence and stability of compensated stochastic theta methods for jump-diffusion SDEs with super-linearly growing coefficients, Calcolo60(2023), no. 2, Paper No. 30, 32. MR 4589707

    2023

  13. [21]

    Wu and X

    X. Wu and X. Wang,Strong convergence rates for long-time approximations of SDEs with non-globally Lipschitz continuous coefficients, arXiv:2406.10582 (2024). 34 ZHIHUI LIU AND XIAOMING WU * Department of Mathematics & National Center for Applied Mathematics Shenzhen (NCAMS) & ...

    2024

Pith tools

Reviewed May 18, 2026 · model on record in the stance chip above.