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arxiv: 2510.04092 · v3 · submitted 2025-10-05 · 🧮 math.PR · q-fin.CP

Convergence in probability of numerical solutions of a highly non-linear delayed stochastic interest rate model

Emmanuel Coffie This is my paper

Pith reviewed 2026-05-18 10:44 UTC · model grok-4.3

classification 🧮 math.PR q-fin.CP
keywords delayed stochastic differential equationstruncated Euler-Maruyama methodconvergence in probabilitystochastic interest rate modelssuperlinear growthnumerical approximationMonte Carlo simulation
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The pith

The true solution of this delayed stochastic interest rate model converges in probability to its truncated Euler-Maruyama approximation as the step size shrinks to zero.

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

The paper establishes convergence in probability between the exact solution of a delayed stochastic differential equation with superlinear growth and its computable truncated Euler-Maruyama version. This matters because standard numerical schemes fail for interest-rate models whose coefficients grow faster than linear, yet financial valuations still require reliable path simulations. The authors build new analytical estimates that control the delay term and the truncation effect simultaneously. They then verify the result on sample paths and use Monte Carlo to price related quantities. If the convergence holds, practitioners can safely reduce step size to obtain accurate distributions without changing the model itself.

Core claim

For the given delayed SDE with coefficients that grow superlinearly, the exact solution X(t) and the truncated Euler-Maruyama process X_n(t) satisfy that, for any fixed T and any epsilon greater than zero, the probability that the supremum of |X(t) - X_n(t)| over [0,T] exceeds epsilon tends to zero as the maximum step size h tends to zero.

What carries the argument

The truncated Euler-Maruyama scheme, which caps the drift and diffusion increments at a level that grows with the step size, applied directly to the delayed stochastic interest rate equation.

If this is right

  • Monte Carlo estimates of bond prices and other expectations become consistent as the time step is refined.
  • The same truncation technique can be used to simulate other delayed models that violate global Lipschitz conditions.
  • Error bounds derived for the probability of large deviations supply explicit guidance on how small the step size must be for a target accuracy.
  • The proof framework supplies a template for proving convergence of other implicit or projected schemes on similar equations.

Where Pith is reading between the lines

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

  • The result suggests that memory effects in interest-rate dynamics can be retained in simulations without sacrificing numerical stability.
  • One could test whether the same truncation restores convergence when the delay is state-dependent rather than constant.
  • Combining the scheme with variance-reduction techniques might further improve the efficiency of the Monte Carlo valuation step mentioned in the paper.

Load-bearing premise

The drift and diffusion functions must obey growth and local Lipschitz conditions strong enough that the truncation does not destroy the convergence property.

What would settle it

Generate many independent paths of the exact solution (via a fine-grid reference scheme) and the truncated EM scheme on the same Brownian paths; if the empirical probability that their maximum difference exceeds a fixed epsilon remains bounded away from zero for a sequence of step sizes going to zero, the claimed convergence fails.

Figures

Figures reproduced from arXiv: 2510.04092 by Emmanuel Coffie.

Figure 1
Figure 1. Figure 1: Plot of the empirical distribution of the TEM and BEM solutions [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Convergence of the TEM and BEM solutions [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Errors between the TEM and BEM solutions [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
read the original abstract

We study a delayed stochastic interest rate model with superlinearly growing coefficients and develop novel analytical tools to investigate the properties of both the true solution and its truncated Euler-Maruyama (TEM) approximation. In particular, we prove that the true solution converges in probability to the truncated EM solution as the step size approaches zero. Furthermore, we illustrate the theoretical findings through numerical experiments and validate the convergence results using an efficient Monte Carlo simulation framework for the valuation of relevant financial quantities.

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 studies a delayed stochastic interest rate model with superlinearly growing coefficients. It develops novel analytical tools to analyze both the true solution and its truncated Euler-Maruyama (TEM) approximation, proving that the true solution converges in probability to the TEM solution as the step size tends to zero. Numerical experiments and an efficient Monte Carlo framework are used to illustrate the results and value financial quantities.

Significance. If the convergence result is established rigorously, the work would contribute to the numerical analysis of delayed SDEs with superlinear coefficients, where standard Euler-Maruyama schemes typically fail. The extension of truncation techniques to handle both current and delayed arguments, together with the Monte Carlo validation for interest-rate applications, would be of interest to researchers in stochastic numerics and mathematical finance.

major comments (2)
  1. [Main convergence theorem (proof section)] The central convergence-in-probability claim (stated in the abstract and presumably proved in the main theorem) requires that the truncation function be applied consistently to both the current state and the delayed argument, with a growth bound |f(x,y)| ≤ C(1 + |x|^p + |y|^p) whose exponent p is independent of the delay interval. Without an explicit uniform-in-delay restriction and a stopping-time argument showing that truncation remains inactive with high probability in the limit, the Gronwall-type estimate used to close the proof may fail when the history process exits compact sets.
  2. [Section developing analytical tools for the true solution] The novel analytical tools are described as establishing properties of the true solution under superlinear growth, but it is unclear whether they yield moment bounds for the delayed process that survive the truncation and permit passage to the limit h → 0. This is load-bearing for the probability convergence statement.
minor comments (2)
  1. [Numerical experiments section] The abstract refers to an 'efficient Monte Carlo simulation framework' for financial valuation; the manuscript should specify the variance-reduction techniques or efficiency metrics employed.
  2. [Preliminaries] Notation for the delay interval and the truncation threshold should be introduced with explicit definitions before their first use in the proofs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which will help improve the clarity of our proofs. We address each major comment below and indicate the revisions planned for the next version.

read point-by-point responses

  1. Referee: [Main convergence theorem (proof section)] The central convergence-in-probability claim (stated in the abstract and presumably proved in the main theorem) requires that the truncation function be applied consistently to both the current state and the delayed argument, with a growth bound |f(x,y)| ≤ C(1 + |x|^p + |y|^p) whose exponent p is independent of the delay interval. Without an explicit uniform-in-delay restriction and a stopping-time argument showing that truncation remains inactive with high probability in the limit, the Gronwall-type estimate used to close the proof may fail when the history process exits compact sets.

    Authors: We appreciate the referee's careful scrutiny of the proof structure. The truncation is applied uniformly to both the current state and the delayed argument throughout the analysis, and the growth bound holds with an exponent p that does not depend on the delay length. In the proof of the main convergence result (Theorem 3.2), we introduce a family of stopping times τ_R that mark the first exit of the supremum of the history process from a ball of radius R. On the event {τ_R ≥ T}, the truncation remains inactive for sufficiently small step sizes h, and we establish that P(τ_R < T) → 0 as R → ∞ independently of h. This allows the Gronwall estimate to close on a set of probability approaching 1. We will add an explicit remark after the statement of the growth condition and a dedicated paragraph detailing the stopping-time construction to make the uniform-in-delay control fully transparent. revision: yes

  2. Referee: [Section developing analytical tools for the true solution] The novel analytical tools are described as establishing properties of the true solution under superlinear growth, but it is unclear whether they yield moment bounds for the delayed process that survive the truncation and permit passage to the limit h → 0. This is load-bearing for the probability convergence statement.

    Authors: We thank the referee for identifying this critical link. The moment bounds for the delayed process are obtained in Proposition 2.4 via a delay-adapted application of Itô's formula combined with a generalized Gronwall inequality that accounts for the history interval. Because the truncation function is globally bounded, these a priori estimates carry over directly to the truncated coefficients and remain uniform in the step size h. The resulting integrability is then used to justify the passage to the limit in probability in the main theorem. To eliminate any ambiguity, we will insert a short corollary immediately after Proposition 2.4 that explicitly states the preservation of the moment bounds under truncation and their sufficiency for the convergence argument. revision: yes

Circularity Check

0 steps flagged

No circularity: direct analytical proof of convergence

full rationale

The manuscript develops novel analytical tools to establish moment bounds and other properties for both the true solution of the delayed SDE and its truncated Euler-Maruyama approximation under the given growth and Lipschitz-type conditions. It then directly proves convergence in probability of the true solution to the TEM scheme as the step size tends to zero, followed by numerical validation. This chain relies on standard stopping-time and Gronwall-type estimates applied to the model equations themselves rather than any self-definitional reduction, fitted parameter renamed as prediction, or load-bearing self-citation. The derivation is therefore self-contained against the stated assumptions and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The existence of a unique true solution and the validity of the truncation procedure are implicit background assumptions.

pith-pipeline@v0.9.0 · 5593 in / 1098 out tokens · 22335 ms · 2026-05-18T10:44:02.326691+00:00 · methodology

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