| Issue |
ESAIM: M2AN
Volume 60, Number 4, July-August 2026
|
|
|---|---|---|
| Page(s) | 1741 - 1768 | |
| DOI | https://doi.org/10.1051/m2an/2026044 | |
| Published online | 03 August 2026 | |
Numerical stability revisited: A family of Benchmark problems for the analysis of explicit stochastic differential equation integrators
1 Warwick Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK.
2 Department of Mathematics & Statistics, University of North Carolina, Charlotte, NC 28223, USA.
* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
9
April
2025
Revised:
23
April
2026
Accepted:
8
May
2026
Abstract
We revisit the numerical stability of four well-established explicit stochastic integration schemes through a new generic benchmark stochastic differential equation (SDE) designed to assess asymptotic statistical accuracy and stability properties. This one-parameter benchmark equation is derived from a general one-dimensional first-order SDE using spatio-temporal nondimensionalization and is employed to evaluate the performance of the (1) Euler–Maruyama, (2) Milstein, (3) Stochastic Heun, and (4) three-stage Runge–Kutta schemes. Our findings reveal that lower-order schemes can outperform higher-order ones over a range of time-step sizes, depending on the benchmark parameters and application context. The theoretical results are validated through a series of numerical experiments, and we discuss their implications for more general applications, including a nonlinear example. Our results suggest that the insights obtained from the linear benchmark problem provide reliable guidance for time-stepping strategies when simulating nonlinear SDEs.
Mathematics Subject Classification: 60H35 / 65L20
Key words: Analysis of explicit numerical integrators for SDEs / asymptotic statistical stability / spatio-temporal nondimensionalization
© The authors. Published by EDP Sciences, SMAI 2026
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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