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Anticipating stochastic Volterra equations

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  • Alòs, Elisa
  • Nualart, David

Abstract

In this paper we establish the existence and uniqueness of a solution for stochastic Volterra equations assuming that the coefficients F(t,s,x) and Gi(t,s,x) are Ft-measurable, for s[less-than-or-equals, slant]t, where {Ft} denotes the filtration generated by the driving Brownian motion. We impose some differentiability assumptions on the coefficients, in the sense of the Malliavin calculus, in the time interval [s,t]. Some properties of the solution are discussed.

Suggested Citation

  • Alòs, Elisa & Nualart, David, 1997. "Anticipating stochastic Volterra equations," Stochastic Processes and their Applications, Elsevier, vol. 72(1), pages 73-95, December.
    • Handle: RePEc:eee:spapps:v:72:y:1997:i:1:p:73-95
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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. David J. Promel & David Scheffels, 2022. "Pathwise uniqueness for singular stochastic Volterra equations with H\"older coefficients," Papers 2212.08029, arXiv.org, revised Jul 2024.
    5. 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).
    6. 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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