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Normal Approximation of Compound Hawkes Functionals

Author

Listed:
  • Mahmoud Khabou

    (Université de Toulouse)

  • Nicolas Privault

    (Nanyang Technological University)

  • Anthony Réveillac

    (Université de Toulouse)

Abstract

We derive quantitative bounds in the Wasserstein distance for the approximation of stochastic integrals with respect to Hawkes processes by a normally distributed random variable. In the case of deterministic and nonnegative integrands, our estimates involve only the third moment of the integrand in addition to a variance term using a squared norm of the integrand. As a consequence, we are able to observe a “third moment phenomenon” in which the vanishing of the first cumulant can lead to faster convergence rates. Our results are also applied to compound Hawkes processes, and improve on the current literature where estimates may not converge to zero in large time or have been obtained only for specific kernels such as the exponential or Erlang kernels.

Suggested Citation

  • Mahmoud Khabou & Nicolas Privault & Anthony Réveillac, 2024. "Normal Approximation of Compound Hawkes Functionals," Journal of Theoretical Probability, Springer, vol. 37(1), pages 549-581, March.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:1:d:10.1007_s10959-022-01233-6
    DOI: 10.1007/s10959-022-01233-6
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    References listed on IDEAS

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    1. Bacry, E. & Delattre, S. & Hoffmann, M. & Muzy, J.F., 2013. "Some limit theorems for Hawkes processes and application to financial statistics," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2475-2499.
    2. Nicolas Privault, 2019. "Third Cumulant Stein Approximation for Poisson Stochastic Integrals," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1461-1481, September.
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