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

Xiaoming Wu; Zhihui Liu

arxiv: 2509.15698 · v2 · submitted 2025-09-19 · 🧮 math.NA · cs.NA· math.PR

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

Zhihui Liu , Xiaoming Wu This is my paper

Pith reviewed 2026-05-18 16:25 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.PR

keywords weak error estimatesexponential ergodicityinvariant measuresbackward Euler methodstochastic theta methodjump-diffusion SODEsmonotone coefficientsdissipative condition

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The pith

Backward Euler method achieves order-one convergence to invariant measures for monotone SODEs without jumps.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

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.

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.

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.

Where Pith is reading between the lines

These are 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.

Load-bearing premise

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

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.

read the original 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}.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

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

The main result is that under a dissipative condition the stochastic theta method with theta greater than 1/2 is exponentially ergodic for monotone jump-diffusion SODEs, and the backward Euler case then yields weak error estimates. In the jump-free setting this produces a time-independent bound that gives first-order convergence of the invariant measures, which closes the question left in their 2025 J. Sci. Comput. paper. The extension to jumps follows the same technical path but is still a non-trivial step forward from the continuous case they already treated. They state the claims precisely and the citation to their earlier work is appropriate rather than circular. The central estimates appear to be fresh derivations rather than reductions to previously fitted constants. The soft spot is the passage from finite-time weak error to the invariant-measure difference. Exponential ergodicity controls each process separately, but the prefactor in the finite-time bound must stay bounded as t goes to infinity for the limit to be time-independent. The dissipative condition gives contraction, yet it is not obvious without the details whether the difference process between exact and numerical solutions admits a uniform Lyapunov or a Gronwall that does not accumulate before the contraction kicks in. If the one-sided Lipschitz constant is not dominated in the right way, the constant could grow with t. This is worth checking in the proofs but does not look like a load-bearing flaw on the abstract alone. The paper is aimed at researchers working on long-time numerical simulation of SDEs with jumps. It is concrete enough and closes a specific gap, so it deserves a serious referee even if revisions are needed on the uniformity argument.

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)

[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)

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.
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

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

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

specific steps

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.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 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.

axioms (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.

pith-pipeline@v0.9.0 · 5641 in / 1382 out tokens · 36585 ms · 2026-05-18T16:25:47.397351+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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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

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work page 2023

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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

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Liu and Z

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work page 2025

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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)

work page arXiv 2024

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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

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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

work page 2011

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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

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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

work page 2024

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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

work page 2005

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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

work page 2023

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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) & Shenzhen International Center for Mathematics, Southern University of Science an...

work page arXiv 2024