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Discrete-time simulation of Stochastic Volterra equations

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  • Richard, Alexandre
  • Tan, Xiaolu
  • Yang, Fan

Abstract

We study discrete-time simulation schemes for stochastic Volterra equations, namely the Euler and Milstein schemes, and the corresponding Multilevel Monte-Carlo method. By using and adapting some results from Zhang (2008), together with the Garsia–Rodemich–Rumsey lemma, we obtain the convergence rates of the Euler scheme and Milstein scheme under the supremum norm. We then apply these schemes to approximate the expectation of functionals of such Volterra equations by the (Multilevel) Monte-Carlo method, and compute their complexity. We finally provide some numerical simulation results.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:141:y:2021:i:c:p:109-138
    DOI: 10.1016/j.spa.2021.07.003
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Ofelia Bonesini & Giorgia Callegaro & Martino Grasselli & Gilles Pag`es, 2023. "From elephant to goldfish (and back): memory in stochastic Volterra processes," Papers 2306.02708, arXiv.org, revised Sep 2023.
    2. Ofelia Bonesini & Antoine Jacquier & Alexandre Pannier, 2023. "Rough volatility, path-dependent PDEs and weak rates of convergence," Papers 2304.03042, arXiv.org.
    3. 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.
    4. 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.
    5. 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.

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