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Existence and uniqueness of solutions to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients

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  • Wang, Zhidong

Abstract

We prove the existence and uniqueness as well as the continuity of the solution to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients. As application, we then study SDEs with fractional integrals.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:78:y:2008:i:9:p:1062-1071
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    References listed on IDEAS

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    1. Cochran, W. George & Lee, Jung-Soon & Potthoff, Jürgen, 1995. "Stochastic Volterra equations with singular kernels," Stochastic Processes and their Applications, Elsevier, vol. 56(2), pages 337-349, April.
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    Cited by:

    1. Aur'elien Alfonsi & Ahmed Kebaier, 2021. "Approximation of Stochastic Volterra Equations with kernels of completely monotone type," Papers 2102.13505, arXiv.org, revised Mar 2022.
    2. Archil Gulisashvili, 2022. "Multivariate Stochastic Volatility Models and Large Deviation Principles," Papers 2203.09015, arXiv.org, revised Nov 2022.
    3. Aur'elien Alfonsi & Guillaume Szulda, 2024. "On non-negative solutions of stochastic Volterra equations with jumps and non-Lipschitz coefficients," Papers 2402.19203, arXiv.org, revised Jul 2024.
    4. Alexandre Pannier & Antoine Jacquier, 2019. "On the uniqueness of solutions of stochastic Volterra equations," Papers 1912.05917, arXiv.org, revised Apr 2020.
    5. Herv'e Andr`es & Benjamin Jourdain, 2024. "Existence, uniqueness and positivity of solutions to the Guyon-Lekeufack path-dependent volatility model with general kernels," Papers 2408.02477, arXiv.org.
    6. Henrique Guerreiro & Jo~ao Guerra, 2022. "VIX pricing in the rBergomi model under a regime switching change of measure," Papers 2201.10391, arXiv.org.
    7. Aur'elien Alfonsi, 2023. "Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation," Papers 2302.07758, arXiv.org, revised Oct 2024.
    8. 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).
    9. Deya, Aurélien & Tindel, Samy, 2011. "Rough Volterra equations 2: Convolutional generalized integrals," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1864-1899, August.
    10. Ackermann, Julia & Kruse, Thomas & Overbeck, Ludger, 2022. "Inhomogeneous affine Volterra processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 250-279.
    11. David Nualart & Bhargobjyoti Saikia, 2023. "Error distribution of the Euler approximation scheme for stochastic Volterra equations," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1829-1876, September.
    12. Richard, Alexandre & Tan, Xiaolu & Yang, Fan, 2021. "Discrete-time simulation of Stochastic Volterra equations," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 109-138.
    13. Jie, Lijuan & Luo, Liangqing & Zhang, Hua, 2024. "One-dimensional McKean–Vlasov stochastic Volterra equations with Hölder diffusion coefficients," Statistics & Probability Letters, Elsevier, vol. 205(C).
    14. Carsten Chong, 2017. "Lévy-driven Volterra Equations in Space and Time," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1014-1058, September.
    15. Prömel, David J. & Scheffels, David, 2023. "Stochastic Volterra equations with Hölder diffusion coefficients," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 291-315.

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    8. Deya, Aurélien & Tindel, Samy, 2011. "Rough Volterra equations 2: Convolutional generalized integrals," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1864-1899, August.
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    10. Aur'elien Alfonsi & Ahmed Kebaier, 2021. "Approximation of Stochastic Volterra Equations with kernels of completely monotone type," Papers 2102.13505, arXiv.org, revised Mar 2022.
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