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A fractional Hawkes process II: Further characterization of the process

Author

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  • Habyarimana, Cassien
  • Aduda, Jane A.
  • Scalas, Enrico
  • Chen, Jing
  • Hawkes, Alan G.
  • Polito, Federico

Abstract

We characterize a Hawkes point process with kernel proportional to the probability density function of Mittag-Leffler random variables. This kernel decays as a power law with exponent β+1∈(1,2]. Several analytical results can be proved, in particular for the expected intensity of the point process and for the expected number of events of the counting process. These analytical results are used to validate algorithms that numerically invert the Laplace transform of the expected intensity as well as Monte Carlo simulations of the process. Finally, Monte Carlo simulations are used to derive the full distribution of the number of events. The algorithms used for this paper are available at https://github.com/habyarimanacassien/Fractional-Hawkes.

Suggested Citation

  • Habyarimana, Cassien & Aduda, Jane A. & Scalas, Enrico & Chen, Jing & Hawkes, Alan G. & Polito, Federico, 2023. "A fractional Hawkes process II: Further characterization of the process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 615(C).
    • Handle: RePEc:eee:phsmap:v:615:y:2023:i:c:s0378437123001516
    DOI: 10.1016/j.physa.2023.128596
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    References listed on IDEAS

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    1. Hainaut, Donatien, 2020. "Fractional Hawkes processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 549(C).
    2. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
    3. Emmanuel Bacry & Iacopo Mastromatteo & Jean-Franc{c}ois Muzy, 2015. "Hawkes processes in finance," Papers 1502.04592, arXiv.org, revised May 2015.
    4. Hainaut, Donatien, 2020. "Fractional Hawkes processes," LIDAM Reprints ISBA 2020009, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Ketelbuters, John-John & Hainaut, Donatien, 2022. "CDS pricing with fractional Hawkes processes," European Journal of Operational Research, Elsevier, vol. 297(3), pages 1139-1150.
    6. Ketelbuters, John John & Hainaut, Donatien, 2022. "CDS Pricing with Fractional Hawkes Processes," LIDAM Reprints ISBA 2022002, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    7. Jang, Jiwook & Oh, Rosy, 2021. "A review on Poisson, Cox, Hawkes, shot-noise Poisson and dynamic contagion process and their compound processes," Annals of Actuarial Science, Cambridge University Press, vol. 15(3), pages 623-644, November.
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    Cited by:

    1. Ulrich Horst & Wei Xu, 2024. "Functional Limit Theorems for Hawkes Processes," Papers 2401.11495, arXiv.org, revised Dec 2024.

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