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arxiv: 2604.21704 · v1 · submitted 2026-04-23 · 🧮 math.NA · cs.NA

Segment convergence for super-linear stochastic functional differential equations by the truncated Euler-Maruyama method

Shounian Deng , Weiyin Fei , Banban Shi This is my paper

Pith reviewed 2026-05-09 21:10 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords stochastic functional differential equationstruncated Euler-Maruyama methodstrong segment convergencesuper-linear growthnumerical approximationmoment boundednessL2-error estimate
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The pith

The truncated Euler-Maruyama scheme delivers strong convergence of numerical segments for SFDEs with super-linear coefficients.

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

This paper moves beyond the usual pointwise convergence results for stochastic functional differential equations and instead establishes convergence of the entire segment process on finite intervals. For equations whose drift and diffusion grow faster than linearly, the authors apply an explicit truncated Euler-Maruyama method that caps the coefficients to restore moment control. They first prove uniform moment bounds for the numerical segment, then bound the L2 gap between its continuous and piecewise-constant versions, and finally obtain a positive strong convergence order. Readers care because segment convergence directly enables numerical study of invariant measures, ergodicity, and path-dependent quantities that arise in finance and other applications.

Core claim

For SFDEs with super-linear drift and diffusion coefficients, the explicit truncated Euler-Maruyama scheme produces a numerical segment process that remains uniformly bounded in moments over any finite time interval, satisfies an L2-error estimate between its continuous and step-process versions, and converges strongly to the true segment process at a positive order.

What carries the argument

The truncated Euler-Maruyama scheme, which applies a truncation function to the drift and diffusion to prevent explosion while remaining fully explicit.

If this is right

  • The numerical segment inherits the ability to approximate invariant measures of the underlying SFDE.
  • Ergodicity of the numerical segment follows from the established strong convergence.
  • The method supplies a practical tool for pricing path-dependent financial options that depend on the full path segment.
  • A supporting numerical example illustrates that the predicted convergence rates are observed in simulation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Segment convergence may allow direct transfer of stability properties from the continuous SFDE to its numerical approximation over longer horizons than the finite-interval proofs cover.
  • The truncation technique could be adapted to other explicit schemes for SFDEs, such as Milstein-type methods, to obtain segment results without losing computational simplicity.
  • In practice, one could test the method on equations with polynomial growth of varying degrees to measure how the observed order depends on the growth exponent.

Load-bearing premise

The drift and diffusion coefficients obey super-linear growth bounds that still allow a suitable truncation level to produce uniform moment bounds for the numerical segment on finite intervals.

What would settle it

Take any concrete super-linear SFDE, fix a truncation level, and compute the L2 distance between the true segment and the numerical segment at a sequence of step sizes going to zero; if this distance fails to decay at the claimed positive order, the convergence result is false.

Figures

Figures reproduced from arXiv: 2604.21704 by Banban Shi, Shounian Deng, Weiyin Fei.

Figure 1
Figure 1. Figure 1: Rate of strong convergence for Example 5.1 [2] Q. Luo, X. Mao, Y. Shen, Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations, Automatica 47 (2011) 2075–2081. [3] J. Bao, G. Yin, C. Yuan, Ergodicity for functional stochastic differential equations and applications, Nonlinear Analysis: Theory, Methods and Applications 98 (2014) 66–82. [4] F. Wu, F. Xi, C.… view at source ↗
read the original abstract

Most existing literature focuses on pointwise convergence (i.e., convergence at a fixed time point) of numerical solutions for Stochastic functional differential equations (SFDEs). In contrast, this paper investigates the strong segment convergence (i.e., the strong order of convergence of the numerical segment process). For SFDEs with super-linear drift and diffusion coefficients, we employ the explicit truncated Euler-Maruyama (EM) scheme. First, we establish the uniform moment boundedness of the truncated EM solution over a finite time interval. Second, we derive the $L^2$-error estimate between the continuous numerical segment and the step numerical segment. Finally, we prove the strong convergence order of the numerical segment generated by the truncated EM. The results can be used to analyze invariant measures and ergodicity of numerical segment, and have important applications in practical problems such as path-dependent financial options. We also provide a numerical example to support the theoretical results.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes strong segment convergence for the truncated Euler-Maruyama method applied to stochastic functional differential equations (SFDEs) with super-linear drift and diffusion coefficients. It proceeds in three steps: proving uniform moment boundedness of the truncated numerical solution over finite time intervals, deriving the L²-error estimate between the continuous numerical segment and the piecewise-constant step segment, and obtaining the strong convergence order of the numerical segment process. The results are illustrated with a numerical example and noted for applications to invariant measures, ergodicity, and path-dependent problems such as financial options.

Significance. If the central claims hold, the work meaningfully extends the literature on numerical methods for SFDEs from pointwise convergence to segment convergence, which is required for path-dependent functionals. The truncated EM approach is a standard technique for super-linear growth, and the explicit moment bounds plus segment error analysis provide a reusable foundation for studying numerical ergodicity. The three-step architecture is clearly delineated and supports reproducibility when the truncation threshold is fixed as described.

major comments (2)
  1. [§3] §3 (moment bounds): the uniform moment estimate for the truncated process requires the truncation threshold to be chosen independently of the step-size h; the manuscript should state explicitly how the threshold function is calibrated to the super-linear growth constants so that the bound remains uniform in h (otherwise the subsequent L² segment error may not pass to the limit).
  2. [§4] §4 (L² segment error): the passage from the continuous numerical segment to the step-function segment invokes a Lipschitz-type estimate on the functional term; the dependence of the constant on the delay length or the modulus of continuity of the functional must be tracked to confirm it does not reduce the final convergence order below the expected rate.
minor comments (2)
  1. [Abstract] The abstract states that a strong convergence order is proved but does not record the precise order (e.g., ½ or 1); adding this information would improve readability.
  2. [Numerical example] In the numerical example, the error tables or plots should include a log-log scale or explicit order computation to allow direct visual verification of the theoretical rate.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We believe the comments can be addressed with minor revisions to enhance clarity, as detailed in our point-by-point responses below.

read point-by-point responses

  1. Referee: [§3] §3 (moment bounds): the uniform moment estimate for the truncated process requires the truncation threshold to be chosen independently of the step-size h; the manuscript should state explicitly how the threshold function is calibrated to the super-linear growth constants so that the bound remains uniform in h (otherwise the subsequent L² segment error may not pass to the limit).

    Authors: We appreciate this comment. Upon re-examination, our truncation threshold is indeed chosen independently of h, calibrated solely based on the super-linear growth constants appearing in the drift and diffusion coefficients and the length of the time interval. Specifically, the threshold R is selected larger than a constant determined by these parameters to ensure the moment bounds are uniform with respect to h. We will revise the manuscript in §3 to include an explicit statement of this calibration procedure, including the dependence on the growth constants, to make the uniformity clear and facilitate the passage to the limit in the error estimates. revision: yes

  2. Referee: [§4] §4 (L² segment error): the passage from the continuous numerical segment to the step-function segment invokes a Lipschitz-type estimate on the functional term; the dependence of the constant on the delay length or the modulus of continuity of the functional must be tracked to confirm it does not reduce the final convergence order below the expected rate.

    Authors: We agree that tracking the dependence is important for rigor. The functional term satisfies a global Lipschitz condition with a constant that depends on the fixed delay length but is independent of the step-size h. Since the delay is fixed and the modulus of continuity is controlled by the Lipschitz property, the constant in the L²-error estimate between the continuous and step segments does not depend on h and thus does not affect the convergence order. We will add a detailed remark in §4 explicitly stating this independence from h and the dependence on the delay length, confirming that the expected strong convergence order is preserved. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard three-step stochastic convergence proof

full rationale

The derivation follows the conventional architecture for truncated Euler-Maruyama on super-linear SFDEs: (1) uniform moment bounds on the truncated process over finite intervals under explicit growth hypotheses, (2) L2 error between continuous and piecewise-constant numerical segments, and (3) passage to the limit yielding segment convergence order. These steps are supported by standard stochastic analysis techniques and concrete truncation functions calibrated to the coefficients, without any reduction to self-defined quantities, fitted inputs renamed as predictions, or load-bearing self-citations. The abstract and described structure confirm the chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard existence/uniqueness assumptions for SFDEs and growth conditions compatible with truncation; no free parameters or invented entities are explicitly introduced in the summary.

free parameters (1)
  • truncation threshold
    The truncation level must be chosen to control super-linear growth while preserving convergence; its selection rule is not specified in the abstract.
axioms (2)
  • domain assumption Existence and uniqueness of solutions to the SFDE under super-linear growth
    Implicitly required for the true solution to exist before comparing to the numerical segment.
  • standard math Standard Ito calculus and stochastic integral properties
    Used throughout the error analysis for SFDEs.

pith-pipeline@v0.9.0 · 5462 in / 1432 out tokens · 47331 ms · 2026-05-09T21:10:17.940059+00:00 · methodology

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Reference graph

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