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<p>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 <i>G</i>-framework. To state more exactly, for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13662_2025_3982_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>p</mi> <mo>≥</mo> <mn>2</mn> </math></EquationSource> <EquationSource Format="TEX">$p \geq 2$</EquationSource> </InlineEquation>, we prove that the <i>p</i>th moment practical exponential stability holds simultaneously for <i>G</i>-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.</p>

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

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  • 发表日期：

    2025年7月30日

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  • Wen Lu

摘要

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 $p \geq 2$ , we prove that the pth 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.