The paper deals with the asymptotical mean-square stability of the linear θ-methods under variable stepsize and transformation approach for stochastic pantograph differential equations. A limiting equation for the analysis of numerical stability is introduced by Kronecker products. Under the condition which guarantee the stability of exact solutions, the optimal stability region of the linear θ-methods under variable stepsize is given by using the limiting equation, i.e. $\theta \in (\frac{1}{2},1]$, which is the same to the deterministic problems. Moreover, the linear θ-methods under the transformation approach are also considered and the result of the stability is improved for $\theta =\frac{1}{2}$. Finally, numerical examples are given to illustrate the asymptotical mean-square stability under variable stepsize and transformation approach.

本文研究了随机缩放微分方程在变步长和变换方法下线性θ方法的渐近均方稳定性。Kronecker 产品引入了数值稳定性分析的极限方程。在保证精确解稳定性的条件下，利用极限方程给出了变步长下线性θ法的最优稳定区域，即$\theta \in (\frac{1}{2},1]$，这与确定性问题相同。此外，还考虑了变换方法下的线性θ方法，提高了结果的稳定性$\theta =\frac{1}{2}$. 最后，给出数值例子说明了变步长和变换方法下的渐近均方稳定性。