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Volume 22, Issue 6
Mean Square Stability of Numerical Method for Stochastic Volterra Integral Equations with Double Weakly Singular Kernels

Omid Farkhondeh Rouz, Sedaghat Shahmorad & Fevzi Erdogan

DOI: 10.4208/ijnam2025-1033

Int. J. Numer. Anal. Mod., 22 (2025), pp. 755-776.

Published online: 2025-08

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  • 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.

  • Keywords

Stochastic Volterra integral equations, weakly singular kernels, stochastic $θ$-scheme, SOE approximation, mean square stability.

  • AMS Subject Headings

45D05, 45G05, 60H20, 65C30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-22-755, author = {Rouz , Omid FarkhondehShahmorad , Sedaghat and Erdogan , Fevzi}, title = {Mean Square Stability of Numerical Method for Stochastic Volterra Integral Equations with Double Weakly Singular Kernels}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2025}, volume = {22}, number = {6}, pages = {755--776}, 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.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2025-1033}, url = {http://global-sci.org/intro/article_detail/ijnam/24292.html} }
TY - JOUR T1 - Mean Square Stability of Numerical Method for Stochastic Volterra Integral Equations with Double Weakly Singular Kernels AU - Rouz , Omid Farkhondeh AU - Shahmorad , Sedaghat AU - Erdogan , Fevzi JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 755 EP - 776 PY - 2025 DA - 2025/08 SN - 22 DO - http://doi.org/10.4208/ijnam2025-1033 UR - https://global-sci.org/intro/article_detail/ijnam/24292.html KW - Stochastic Volterra integral equations, weakly singular kernels, stochastic $θ$-scheme, SOE approximation, mean square stability. AB -

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.

Rouz , Omid FarkhondehShahmorad , Sedaghat and Erdogan , Fevzi. (2025). Mean Square Stability of Numerical Method for Stochastic Volterra Integral Equations with Double Weakly Singular Kernels. International Journal of Numerical Analysis and Modeling. 22 (6). 755-776. doi:10.4208/ijnam2025-1033
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