A note on some stability results for stochastic delay differential equations

Yao Lu; Dehao Ruan; Quanxin Zhu; Yao Lu; Dehao Ruan; Quanxin Zhu

doi:10.3934/math.20251122

AIMS Mathematics

2025, Volume 10, Issue 11: 25346-25357. doi: 10.3934/math.20251122

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A note on some stability results for stochastic delay differential equations

Yao Lu ^1,2,
Dehao Ruan ^{1,2
,
,},
Quanxin Zhu ^{2
,
,}

1.
School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou 510665, China
2.
CHP-LCOCS, School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan, 410081, China

Received: 06 September 2025 Revised: 19 October 2025 Accepted: 23 October 2025 Published: 04 November 2025
MSC : 34K50, 90B15, 93D20

We investigate the stability of stochastic delay differential equations (SDDEs) that describe systems affected by both memory effects and stochastic disturbances. By adopting a comparison principle approach, we derive unified criteria for $ p $-th moment, asymptotic, and exponential stability under conditions that relax the conventional requirement of a negative diffusion operator. The proposed results broaden the applicability of classical methods and are validated through illustrative numerical examples, demonstrating their potential relevance to engineering and applied sciences.
- stability,
- stochastic delay differential equations,
- Halanay inequalities
Citation: Yao Lu, Dehao Ruan, Quanxin Zhu. A note on some stability results for stochastic delay differential equations[J]. AIMS Mathematics, 2025, 10(11): 25346-25357. doi: 10.3934/math.20251122

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Abstract

We investigate the stability of stochastic delay differential equations (SDDEs) that describe systems affected by both memory effects and stochastic disturbances. By adopting a comparison principle approach, we derive unified criteria for $ p $-th moment, asymptotic, and exponential stability under conditions that relax the conventional requirement of a negative diffusion operator. The proposed results broaden the applicability of classical methods and are validated through illustrative numerical examples, demonstrating their potential relevance to engineering and applied sciences.

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