Mean Square Stability of Numerical Method for Stochastic Volterra Integral Equations with Double Weakly Singular Kernels

Abstract

The main goal of this paper is to develop an improved stochastic $\theta$-scheme as a numerical method for stochastic Volterra integral equations (SVIEs) with double weakly singular kernels and demonstrate that the stability of the proposed scheme is affected by the kernel parameters. To overcome the low computational efficiency of the stochastic $\theta$-scheme, we employed the sum-of-exponentials (SOE) approximation. Then, the mean square stability of the proposed scheme with respect to a convolution test equation is studied. Additionally, based on the stability conditions and the explicit structure of the stability matrices, analytical and numerical stability regions are plotted and compared with the split-step $\theta$-method and the $\theta$-Milstein method. The results confirm that our approach aligns significantly with the expected physical interpretations.