This paper focuses on investigating the stochastic Volterra integral equations (SVIEs) with double weakly singular kernels. Our primary objective is to examine the well-posedness of the proposed equation. Specifically, we explore the presence of existence, uniqueness, boundedness, and the continuous dependence of the exact solution on the initial data. Additionally, we develop a stochastic -scheme as a numerical solution for the equation and demonstrate that the convergence rate of the scheme is influenced by the kernel parameters. To validate the accuracy and reliability of our theoretical findings, we present two numerical examples. Notably, one of these examples concentrates on estimating the price of a European call option using the Heston stochastic volatility model with a singular kernel. Our results, when compared to the corresponding findings by Li et al. , not only relax the integrable limitations of singular kernels but also establish a precise convergence order. In addition, we propose an improved scheme, based on the efficient sum-of-exponentials (SOE) approximation, to address the low computational efficiency of the stochastic -scheme. The results confirm that our approach aligns significantly with the expected physical interpretations.

中文翻译：

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随机Volterra积分方程中的双弱奇异核及其在粗糙Heston模型中的应用

本文重点研究具有双弱奇异核的随机 Volterra 积分方程 (SVIE)。我们的主要目标是检查所提出方程的适定性。具体来说，我们探索存在性、唯一性、有界性以及精确解对初始数据的持续依赖。此外，我们开发了一个随机方案作为方程的数值解，并证明该方案的收敛速度受到核参数的影响。为了验证我们的理论研究结果的准确性和可靠性，我们提供了两个数值示例。值得注意的是，其中一个示例集中于使用具有奇异内核的赫斯顿随机波动率模型来估计欧式看涨期权的价格。我们的结果与 Li 等人的相应发现进行了比较。 ，不仅放宽了奇异核的可积限制，而且建立了精确的收敛阶。此外，我们提出了一种基于高效指数和（SOE）近似的改进方案，以解决随机方案计算效率低的问题。结果证实我们的方法与预期的物理解释显着一致。