In this paper, a general theorem on the equivalence of pth moment stability between stochastic differential delay equations (SDDEs) and their numerical methods is proved under the assumptions that the numerical methods are strongly convergent and have the bouneded $p$th moment in the finite time. The truncated Euler-Maruyama (EM) method is studied as an example to illustrate that the theorem indeed covers a large ranges of SDDEs. Alongside the investigation of the truncated EM method, the requirements on the step size of the method are significantly released compared with the work, where the method was initially proposed.

中文翻译：

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随机微分时滞方程及其数值方法的第p阶矩稳定性的等价性

在本文中，在数值方法强收敛且在有限元中具有有界$p$th矩的假设下，证明了随机微分延迟方程（SDDE）与其数值方法之间p阶矩稳定性等价的一般定理。时间。以截断的 Euler-Maruyama (EM) 方法为例，说明该定理确实涵盖了大范围的 SDDE。除了对截断 EM 方法的研究之外，与最初提出该方法的工作相比，对方法步长的要求也得到了显着释放。