In this paper, a split-step $\theta$ method is introduced for solving stochastic Volterra integral equations with general smooth kernels. First, the mean-square boundedness and convergence properties of the numerical solution are analyzed. In particular, when the kernel function in the stochastic integral term satisfies a certain condition, the method can achieve superconvergence. Then, the mean-square stability of the method with respect to a convolution test equation is studied. A recurrence relation is found and mean-square stability regions are given by using root locus method. In particular, when the test equation degrades to the deterministic case, the new method with $\theta \ge 1∕2$ is $V_0$-stable (i.e., it can preserve the stability of the convolution test equation), which is superior to the stochastic $\theta$ method. Finally, some numerical experiments are given to verify the theoretical results.

本文分步进行 $\theta$介绍了用一般光滑核求解随机Volterra积分方程的方法。首先，分析了数值解的均方有界性和收敛性。特别地，当随机积分项中的核函数满足一定条件时，该方法可以实现超收敛。然后，研究了该方法相对于卷积测试方程的均方稳定性。通过使用根轨迹法找到了递归关系，并给出了均方稳定区域。特别是当测试方程退化为确定性情况时，新方法具有$\theta \ge \mathrm{1个}∕2$ 是 $V_0$-稳定（即可以保留卷积测试方程的稳定性），优于随机 $\theta$方法。最后，通过数值实验验证了理论结果。