This paper focuses on explicit approximations for nonlinear stochastic delay differential equations (SDDEs). Under less restrictive conditions, the truncated Euler-Maruyama (TEM) schemes for SDDEs are proposed, which numerical solutions are bounded in the q th moment for q ≥ 2 and converge to the exact solutions strongly in any finite interval. The 1/2 order convergence rate is yielded. Furthermore, the long-time asymptotic behaviors of numerical solutions, such as stability in mean square and \(\mathbb {P}-1\), are examined. Several numerical experiments are carried out to illustrate our results.

中文翻译：

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非线性随机延迟微分方程显式逼近的强收敛性和稳定性

本文重点介绍非线性随机延迟微分方程 (SDDE) 的显式近似。在限制较少的条件下，提出了 SDDE 的截断 Euler-Maruyama (TEM) 方案，其数值解在q ≥ 2的q时刻有界，并在任何有限区间内强烈收敛到精确解。产生了 1/2 阶收敛率。此外，还检查了数值解的长期渐近行为，例如均方稳定性和\(\mathbb {P}-1\)。进行了几个数值实验来说明我们的结果。