Normal Approximation of Compound Hawkes Functionals

Mahmoud Khabou (), Nicolas Privault and Anthony Réveillac ()
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Mahmoud Khabou: Université de Toulouse
Anthony Réveillac: Université de Toulouse

Journal of Theoretical Probability, 2024, vol. 37, issue 1, 549-581

Abstract: 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.

Keywords: Hawkes processes; Stein’s method; Normal approximation; Malliavin calculus; 60G55; 60F05; 60G57; 60H07 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-022-01233-6

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