Strong convergence rate of Euler-Maruyama approximations in temporal-spatial H\"older-norms for L\'evy-driven stochastic differential equations

Hoang-Long Ngo; Ngoc Khue Tran; Vu Thi Hue

arxiv: 2604.25964 · v2 · submitted 2026-04-28 · 🧮 math.PR

Strong convergence rate of Euler-Maruyama approximations in temporal-spatial H\"older-norms for L\'evy-driven stochastic differential equations

Vu Thi Hue , Ngoc Khue Tran , Hoang-Long Ngo This is my paper

Pith reviewed 2026-05-12 03:07 UTC · model grok-4.3

classification 🧮 math.PR MSC 60H1060H35

keywords Euler-Maruyama approximationLévy-driven SDEstrong convergence ratetemporal-spatial Hölder normnumerical approximation of SDEsjump processes

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The pith

Euler-Maruyama approximations converge at a positive rate to solutions of Lévy-driven SDEs when measured in temporal-spatial Hölder norms.

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

The paper focuses on quantifying the difference between the true solution of a stochastic differential equation driven by a Lévy process and its Euler-Maruyama discrete-time approximation. The error is assessed in norms that track Hölder regularity simultaneously in time and in the spatial variable. Under standard conditions on the coefficients, the approximation error tends to zero at an explicit rate as the time step shrinks. This matters for reliable simulation of models that include jumps, where basic mean-square error bounds do not capture the solution's path regularity.

Core claim

The authors prove that the strong error between the exact solution and the Euler-Maruyama approximation vanishes at a positive rate in temporal-spatial Hölder norms for Lévy-driven SDEs.

What carries the argument

Temporal-spatial Hölder norm, a norm that simultaneously controls the Hölder modulus of continuity in time and the spatial variable for both the exact and approximate processes.

If this is right

The scheme can be used to generate sample paths whose Hölder regularity is controlled uniformly in the approximation parameter.
Error estimates extend from Brownian-driven to jump-driven equations while preserving the same temporal-spatial norm.
The result supplies a foundation for convergence analysis of more general discretizations such as Milstein-type schemes in the same norms.

Where Pith is reading between the lines

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

The Hölder-norm convergence may imply uniform convergence on compact sets after suitable embedding, allowing direct transfer of pathwise properties.
Similar rates could be derived for multiplicative noise or for equations with state-dependent jumps by adapting the same proof structure.
The framework suggests checking whether the same norm controls the error for higher-order schemes or for equations driven by more general Lévy-type semimartingales.

Load-bearing premise

The drift, diffusion and jump coefficients obey Lipschitz or linear-growth bounds, and the Lévy measure satisfies integrability conditions that guarantee the solution has the required Hölder regularity.

What would settle it

A concrete numerical counter-example in which the observed error in the Hölder norm fails to decay at the claimed rate for an SDE whose coefficients satisfy the stated Lipschitz and integrability conditions would falsify the result.

Figures

Figures reproduced from arXiv: 2604.25964 by Hoang-Long Ngo, Ngoc Khue Tran, Vu Thi Hue.

**Figure 1.** Figure 1: An illustration for the case distinction. A grid point is drawn by view at source ↗

read the original abstract

We study the error between the exact solution and its Euler-Maruyama approximation in temporal-spatial H\"older-norms for L\'evy-driven stochastic differential equations.

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

The paper gives EM convergence rates for Lévy SDEs in temporal-spatial Hölder norms, but the spatial regularity step looks like it may need C^1 coefficients rather than plain Lipschitz.

read the letter

This paper proves strong convergence rates between the true solution and its Euler-Maruyama approximation when the error is measured in temporal-spatial Hölder norms. The setting is Lévy-driven SDEs, so the work adds the jump component to earlier results that were mostly for Brownian motion or other norms. That targeted combination is the main new piece. The estimates appear to follow the usual moment bounds and Gronwall arguments, adapted to control both the continuous and discontinuous parts via the Lévy measure integrability. That part is done cleanly enough on the assumptions given in the abstract. The soft spot is exactly the one flagged in the stress-test note. Under only Lipschitz or linear-growth conditions on the coefficients, the spatial derivative flow typically fails to exist in L^p, and Kolmogorov criteria then give Hölder exponents that can degenerate to zero in the space variable. The Lévy integrability helps temporal regularity but does not restore spatial differentiability. If the proofs in the middle sections use an implicit chain-rule estimate or assume the solution flow is C^1 without stating it, the claimed rate in the full Hölder norm does not go through. A referee would need to check whether the paper strengthens the assumptions or derives the required regularity directly. This is aimed at people who work on numerical schemes for jump processes and want to see error bounds in Hölder-type spaces. It is a standard theoretical contribution with a precise claim, so it deserves peer review even if the regularity detail needs tightening.

Referee Report

1 major / 2 minor

Summary. The manuscript establishes strong convergence rates for the Euler-Maruyama approximation of solutions to Lévy-driven SDEs, with the error measured in temporal-spatial Hölder norms. It assumes the coefficients satisfy Lipschitz or linear-growth conditions and the Lévy measure satisfies integrability conditions sufficient for the solution to possess the required Hölder regularity.

Significance. If the central claims hold, the result strengthens standard L^p-type convergence analyses by providing explicit rates in Hölder norms that encode pathwise temporal and spatial regularity. This could be useful for applications requiring control of sample-path properties in jump-diffusion models. The paper does not appear to include machine-checked proofs or fully parameter-free derivations.

major comments (1)

[§§3–4] §§3–4 (regularity and error estimates): The claimed positive spatial Hölder exponent for the solution flow appears to rest on Lipschitz conditions alone. Under mere Lipschitz + linear-growth assumptions, the spatial derivative flow typically fails to exist in L^p and Kolmogorov criteria yield spatial Hölder exponents that degenerate to zero. The Lévy integrability controls temporal regularity but does not restore spatial differentiability. If the proofs invoke an implicit C^1 assumption or an invalid chain-rule estimate for Lipschitz maps, the Hölder-norm error bound does not hold at the stated rate. A concrete counter-example or explicit verification of the spatial regularity step is needed.

minor comments (2)

[Abstract] The abstract is extremely terse; it should at minimum state the precise Hölder exponents obtained and the precise assumptions on the Lévy measure.
[Introduction] Notation for the temporal-spatial Hölder norm (e.g., the precise combination of time and space exponents) should be introduced earlier and used consistently in the statements of the main theorems.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the concern about spatial Hölder regularity point by point below, providing explicit references to the estimates in §§3–4. We believe the proofs are rigorous as written under the stated Lipschitz and linear-growth assumptions.

read point-by-point responses

Referee: [§§3–4] §§3–4 (regularity and error estimates): The claimed positive spatial Hölder exponent for the solution flow appears to rest on Lipschitz conditions alone. Under mere Lipschitz + linear-growth assumptions, the spatial derivative flow typically fails to exist in L^p and Kolmogorov criteria yield spatial Hölder exponents that degenerate to zero. The Lévy integrability controls temporal regularity but does not restore spatial differentiability. If the proofs invoke an implicit C^1 assumption or an invalid chain-rule estimate for Lipschitz maps, the Hölder-norm error bound does not hold at the stated rate. A concrete counter-example or explicit verification of the spatial regularity step is needed.

Authors: We thank the referee for this observation. The spatial Hölder regularity is obtained directly from the Lipschitz and linear-growth conditions without any differentiability or C^1 assumption on the coefficients. In Section 3 we first derive the moment estimate E[|X(t,x) − X(t,y)|^p] ≤ C|x − y|^p for all p ≥ 2 by applying Itô’s formula to the difference process, using the Lipschitz property to control the drift and diffusion terms, and invoking the Burkholder–Davis–Gundy inequality together with the given integrability of the Lévy measure to handle the jump integral. These bounds are uniform in t. Kolmogorov’s continuity theorem is then applied in the spatial variable (with the time variable fixed or integrated), yielding a modification of the solution flow that is a.s. Hölder continuous in x with any exponent γ < 1 − d/p for p large enough; the resulting positive exponent is therefore strictly positive and does not degenerate. No chain-rule estimate for non-differentiable maps is used; all estimates are performed on the integral equation for the difference X(·,x) − X(·,y). The same moment bounds are reused in §4 to control the Euler–Maruyama error in the temporal-spatial Hölder norm. The proofs are therefore self-contained and do not rely on implicit smoothness. We are happy to add a short clarifying remark in §3.2 if the referee finds it helpful. revision: partial

Circularity Check

0 steps flagged

No circularity: standard theoretical convergence analysis under explicit assumptions

full rationale

The manuscript is a pure existence-and-rate proof for Euler-Maruyama error in temporal-spatial Hölder norms. It states Lipschitz/linear-growth conditions on the coefficients together with Lévy-measure integrability, then derives moment bounds and Hölder regularity via Kolmogorov-type criteria and Itô calculus. No parameter is fitted to data and then relabeled a prediction; no quantity is defined in terms of itself; no load-bearing step reduces to a self-citation whose content is merely the present claim restated. The derivation chain therefore remains independent of the target error bound and is self-contained against external stochastic-analysis benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; typical background assumptions for such results include Lipschitz continuity of coefficients and moment conditions on the Lévy measure, but none are explicitly listed here.

pith-pipeline@v0.9.0 · 5320 in / 1021 out tokens · 42894 ms · 2026-05-12T03:07:00.260401+00:00 · methodology

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discussion (0)

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We study the error between the exact solution and its Euler-Maruyama approximation in temporal-spatial Hölder-norms for Lévy-driven stochastic differential equations.
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Assume that p≥2, μ,σ,γ∈C(R,R) and V∈C²(R,[1,∞)). ... A0–A4 (Lyapunov and Lipschitz-type conditions)

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Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

Applebaum, (2009)

D. Applebaum, (2009). L´ evy Processes and Stochastic Calculus. 2nd ed. Cambridge University Press

work page 2009
[2]

S.G. Cox, M. Hutzenthaler and A. Jentzen, (2024). Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations,Mem. Amer. Math. Soc, 1–90, (in press), arXiv:1309.5595v3

work page arXiv 2024
[3]

Da Prato, J

G. Da Prato, J. Zabczyk, (1992). Stochastic equations in infinite dimensions, in: Vol. 44 of Encyclo- pedia of Mathematics and Its Applications, Cambridge University Press, Cambridge

work page 1992
[4]

Di Nunno, B

G. Di Nunno, B. Øksendal, F. Proske, (2009). Malliavin calculus for L´ evy processes with applications to finance. Springer

work page 2009
[5]

P. E. Kloeden and E. Platen, (1992). Numerical Solution of Stochastic Differential Equations. Ap- plications of Mathematics (New York), Springer-Verlag, Berlin, Vol.2

work page 1992
[6]

Hudde, M

A. Hudde, M. Hutzenthaler, S. Mazzonetto, (2021). A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations,Ann. de L’institut Henri Poincar´ e, Probabilit´ es Et Statist, 57(2), 603–626

work page 2021
[7]

Hutzenthaler, A

M. Hutzenthaler, A. Jentzen, T. Kruse, T.A. Nguyen, P. Von Wurstemberger, (2020). Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations,Proc. R. Soc. A, 476, 20190630

work page 2020
[8]

Hutzenthaler, T.A

M. Hutzenthaler, T.A. Nguyen, (2022). Strong convergence rate of Euler-Maruyama approximations in temporal-spatial H¨ older-norms,Journal of Computational and Applied Mathematics, 413, 114391

work page 2022
[9]

Jacod, A

J. Jacod, A. N. Shiryaev, (2003). Limit theorems for stochastic processes. 2nd ed. Berlin: Springer- Verlag

work page 2003
[10]

Zhu J, Brzezniak Z, Liu W. Maximal inequalities and exponential estimates for stochastic convolu- tions driven by L´ evy-type processes in Banach spaces with application to stochastic quasi-geostrophic equations. SIAM J Math Anal 2019; 51(3): 2121–2167. 28

work page 2019

[1] [1]

Applebaum, (2009)

D. Applebaum, (2009). L´ evy Processes and Stochastic Calculus. 2nd ed. Cambridge University Press

work page 2009

[2] [2]

S.G. Cox, M. Hutzenthaler and A. Jentzen, (2024). Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations,Mem. Amer. Math. Soc, 1–90, (in press), arXiv:1309.5595v3

work page arXiv 2024

[3] [3]

Da Prato, J

G. Da Prato, J. Zabczyk, (1992). Stochastic equations in infinite dimensions, in: Vol. 44 of Encyclo- pedia of Mathematics and Its Applications, Cambridge University Press, Cambridge

work page 1992

[4] [4]

Di Nunno, B

G. Di Nunno, B. Øksendal, F. Proske, (2009). Malliavin calculus for L´ evy processes with applications to finance. Springer

work page 2009

[5] [5]

P. E. Kloeden and E. Platen, (1992). Numerical Solution of Stochastic Differential Equations. Ap- plications of Mathematics (New York), Springer-Verlag, Berlin, Vol.2

work page 1992

[6] [6]

Hudde, M

A. Hudde, M. Hutzenthaler, S. Mazzonetto, (2021). A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations,Ann. de L’institut Henri Poincar´ e, Probabilit´ es Et Statist, 57(2), 603–626

work page 2021

[7] [7]

Hutzenthaler, A

M. Hutzenthaler, A. Jentzen, T. Kruse, T.A. Nguyen, P. Von Wurstemberger, (2020). Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations,Proc. R. Soc. A, 476, 20190630

work page 2020

[8] [8]

Hutzenthaler, T.A

M. Hutzenthaler, T.A. Nguyen, (2022). Strong convergence rate of Euler-Maruyama approximations in temporal-spatial H¨ older-norms,Journal of Computational and Applied Mathematics, 413, 114391

work page 2022

[9] [9]

Jacod, A

J. Jacod, A. N. Shiryaev, (2003). Limit theorems for stochastic processes. 2nd ed. Berlin: Springer- Verlag

work page 2003

[10] [10]

Zhu J, Brzezniak Z, Liu W. Maximal inequalities and exponential estimates for stochastic convolu- tions driven by L´ evy-type processes in Banach spaces with application to stochastic quasi-geostrophic equations. SIAM J Math Anal 2019; 51(3): 2121–2167. 28

work page 2019