| Issue |
ESAIM: M2AN
Volume 59, Number 4, July-August 2025
|
|
|---|---|---|
| Page(s) | 2207 - 2251 | |
| DOI | https://doi.org/10.1051/m2an/2025052 | |
| Published online | 31 July 2025 | |
Uniform in time convergence of numerical schemes for stochastic differential equations via strong exponential stability: Euler methods, split-step and tamed schemes
1
Maxwell Institute for Mathematical Sciences and Mathematics Department, Heriot-Watt University, Edinburgh EH14 4AS, UK
2
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
* Corresponding author: m.ottobre@hw.ac.uk
Received:
26
July
2024
Accepted:
16
June
2025
We prove a general criterion providing sufficient conditions under which a time-discretiziation of a given Stochastic Differential Equation (SDE) is a uniform in time approximation of the SDE. The criterion is also, to a certain extent, discussed in the paper, necessary. Using such a criterion we then analyse the convergence properties of numerical methods for solutions of SDEs; we consider Explicit and Implicit Euler, split-step and (truncated) tamed Euler methods. In particular, we show that, under mild conditions on the coefficients of the SDE (locally Lipschitz and strictly monotone), these methods produce approximations of the law of the solution of the SDE that converge uniformly in time. The bounds we provide are non-asymptotic. The theoretical results are verified by numerical examples.
Mathematics Subject Classification: 65C20 / 65C30 / 60H10 / 65G99 / 47D07 / 60J60
Key words: Stochastic Differential Equations / numerical methods for SDEs / Explicit and Implicit Euler schemes / split step / tamed Euler schemes / derivative estimates / Markov semigroups / strong exponential stability / uniform in time bounds
© The authors. Published by EDP Sciences, SMAI 2025
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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