Practical stability equivalence for SDEs, SDDEs, and their associated Euler-Maruyama methods in the G-framework

• Published in: Advances in continuous and discrete models Vol. 2025; no. 1; p. 121
• Main Author: Lu, Wen
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 30.07.2025; Springer Nature B.V
• Subjects: Analysis; Approximation; Brownian motion; Difference and Functional Equations; Differential equations; Equivalence; Functional Analysis; Inequality; Mathematics; Mathematics and Statistics; Methods; Numerical analysis; Ordinary Differential Equations; Partial Differential Equations; Stability; Time lag; Brownian motion; Euler-Maruyama method; Stochastic differential equations; Equivalence; Practical exponential stability
• ISSN: 2731-4235; 1687-1839; 2731-4235; 1687-1847
• DOI: 10.1186/s13662-025-03982-0

This paper aims to study the equivalence of stability between stochastic differential equations with time delay (SDDEs), their auxiliary stochastic differential equations (SDEs), and the associated Euler-Maruyama (EM) methods under the G -framework. To state more exactly, for p ≥ 2 , we prove that the p th moment practical exponential stability holds simultaneously for G -Brownian motion driven SDDEs (GSDDEs), their auxiliary SDEs (GSDEs), and the associated EM methods for both GSDDEs and GSDEs, when either the time delay or step size is sufficiently small. A numerical example demonstrates this equivalence.