The mean-square stability of the $\theta$-method for neutral stochastic delay integro-differential equations (NSDIDEs) is considered in this paper. We construct two classes of $\theta$-methods, i.e. the stochastic linear theta (SLT) method and the split-step theta (SST) method for NSDIDEs. Under the one-sided growth condition and contractive condition, we show that both methods are mean-square exponentially stable. An example is given to illustrate the theoretical results.

中文翻译：

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两类均值的均方稳定性 $\theta$随机延迟积分微分方程的数学方法

的均方稳定性 $\theta$本文考虑了中立随机延迟积分微分方程（NSDIDE）的方法。我们构造了两类$\theta$-方法，即用于NSDIDE的随机线性theta（SLT）方法和分段式theta（SST）方法。在单边生长条件和收缩条件下，我们证明这两种方法都是均方指数稳定的。举例说明了理论结果。