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Double weakly singular kernels in stochastic Volterra integral equations with application to the rough Heston model

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  • Farkhondeh Rouz, O.
  • Shahmorad, S.
  • Ahmadian, D.

Abstract

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 θ-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. [3], 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 θ-scheme. The results confirm that our approach aligns significantly with the expected physical interpretations.

Suggested Citation

  • Farkhondeh Rouz, O. & Shahmorad, S. & Ahmadian, D., 2024. "Double weakly singular kernels in stochastic Volterra integral equations with application to the rough Heston model," Applied Mathematics and Computation, Elsevier, vol. 475(C).
    • Handle: RePEc:eee:apmaco:v:475:y:2024:i:c:s0096300324001929
    DOI: 10.1016/j.amc.2024.128720
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    References listed on IDEAS

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    1. Wang, Zhidong, 2008. "Existence and uniqueness of solutions to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients," Statistics & Probability Letters, Elsevier, vol. 78(9), pages 1062-1071, July.
    2. Siow Woon Jeng & Adem Kiliçman, 2021. "On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model," Mathematics, MDPI, vol. 9(22), pages 1-32, November.
    3. Zhao, Qian & Wang, Rongming & Wei, Jiaqin, 2016. "Exponential utility maximization for an insurer with time-inconsistent preferences," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 89-104.
    4. Nacira Agram & Bernt Øksendal, 2015. "Malliavin Calculus and Optimal Control of Stochastic Volterra Equations," Journal of Optimization Theory and Applications, Springer, vol. 167(3), pages 1070-1094, December.
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