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Hawkes Processes Framework With a Gamma Density As Excitation Function: Application to Natural Disasters for Insurance

Author

Listed:
  • Laurent Lesage

    (University of Lorraine, CNRS, Inria, IECL
    University of Luxembourg)

  • Madalina Deaconu

    (University of Lorraine, CNRS, Inria, IECL)

  • Antoine Lejay

    (University of Lorraine, CNRS, Inria, IECL)

  • Jorge Augusto Meira

    (University of Luxembourg)

  • Geoffrey Nichil

    (Foyer Assurances)

  • Radu State

    (University of Luxembourg)

Abstract

Hawkes processes are temporal self-exciting point processes. They are well established in earthquake modelling or finance and their application is spreading to diverse areas. Most models from the literature have two major drawbacks regarding their potential application to insurance. First, they use an exponentially-decaying form of excitation, which does not allow a delay between the occurrence of an event and its excitation effect on the process and does not fit well on insurance data consequently. Second, theoretical results developed from these models are valid only when time of observation tends to infinity, whereas the time horizon for an insurance use case is of several months or years. In this paper, we define a complete framework of Hawkes processes with a Gamma density excitation function (i.e. estimation, simulation, goodness-of-fit) instead of an exponential-decaying function and we demonstrate some mathematical properties (i.e. expectation, variance) about the transient regime of the process. We illustrate our results with real insurance data about natural disasters in Luxembourg.

Suggested Citation

  • Laurent Lesage & Madalina Deaconu & Antoine Lejay & Jorge Augusto Meira & Geoffrey Nichil & Radu State, 2022. "Hawkes Processes Framework With a Gamma Density As Excitation Function: Application to Natural Disasters for Insurance," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2509-2537, December.
    • Handle: RePEc:spr:metcap:v:24:y:2022:i:4:d:10.1007_s11009-022-09938-1
    DOI: 10.1007/s11009-022-09938-1
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    References listed on IDEAS

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    1. Peter Halpin & Paul Boeck, 2013. "Modelling Dyadic Interaction with Hawkes Processes," Psychometrika, Springer;The Psychometric Society, vol. 78(4), pages 793-814, October.
    2. Mohler, G. O. & Short, M. B. & Brantingham, P. J. & Schoenberg, F. P. & Tita, G. E., 2011. "Self-Exciting Point Process Modeling of Crime," Journal of the American Statistical Association, American Statistical Association, vol. 106(493), pages 100-108.
    3. Xuefeng Gao & Lingjiong Zhu, 2018. "Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues," Queueing Systems: Theory and Applications, Springer, vol. 90(1), pages 161-206, October.
    4. Zailei Cheng & Youngsoo Seol, 2020. "Diffusion Approximation of a Risk Model with Non-Stationary Hawkes Arrivals of Claims," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 555-571, June.
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    Cited by:

    1. Mercuri, Lorenzo & Perchiazzo, Andrea & Rroji, Edit, 2024. "A Hawkes model with CARMA(p,q) intensity," Insurance: Mathematics and Economics, Elsevier, vol. 116(C), pages 1-26.
    2. Kyungsub Lee, 2024. "Discrete Hawkes process with flexible residual distribution and filtered historical simulation," Papers 2401.13890, arXiv.org.

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