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Hawkes processes framework with a Gamma density as excitation function: application to natural disasters for insurance

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  • Laurent Lesage

    (IECL - Institut Élie Cartan de Lorraine - UL - Université de Lorraine - CNRS - Centre National de la Recherche Scientifique, SnT - Interdisciplinary Centre for Security, Reliability and Trust [Luxembourg] - uni.lu - Université du Luxembourg = University of Luxembourg = Universität Luxemburg, Foyer Assurances [Leudelange], PASTA - Processus aléatoires spatio-temporels et leurs applications - Inria Nancy - Grand Est - Inria - Institut National de Recherche en Informatique et en Automatique - IECL - Institut Élie Cartan de Lorraine - UL - Université de Lorraine - CNRS - Centre National de la Recherche Scientifique)

  • Madalina Deaconu

    (IECL - Institut Élie Cartan de Lorraine - UL - Université de Lorraine - CNRS - Centre National de la Recherche Scientifique, PASTA - Processus aléatoires spatio-temporels et leurs applications - Inria Nancy - Grand Est - Inria - Institut National de Recherche en Informatique et en Automatique - IECL - Institut Élie Cartan de Lorraine - UL - Université de Lorraine - CNRS - Centre National de la Recherche Scientifique)

  • Antoine Lejay

    (IECL - Institut Élie Cartan de Lorraine - UL - Université de Lorraine - CNRS - Centre National de la Recherche Scientifique, PASTA - Processus aléatoires spatio-temporels et leurs applications - Inria Nancy - Grand Est - Inria - Institut National de Recherche en Informatique et en Automatique - IECL - Institut Élie Cartan de Lorraine - UL - Université de Lorraine - CNRS - Centre National de la Recherche Scientifique)

  • Jorge Augusto Meira

    (SnT - Interdisciplinary Centre for Security, Reliability and Trust [Luxembourg] - uni.lu - Université du Luxembourg = University of Luxembourg = Universität Luxemburg)

  • Geoffrey Nichil

    (Foyer Assurances [Leudelange])

  • Radu State

    (SnT - Interdisciplinary Centre for Security, Reliability and Trust [Luxembourg] - uni.lu - Université du Luxembourg = University of Luxembourg = Universität Luxemburg)

Abstract

Hawkes process 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," Post-Print hal-03040090, HAL.
  • Handle: RePEc:hal:journl:hal-03040090
    DOI: 10.1007/s11009-022-09938-1
    Note: View the original document on HAL open archive server: https://inria.hal.science/hal-03040090v1
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    References listed on IDEAS

    as
    1. Emmanuel Bacry & Iacopo Mastromatteo & Jean-Franc{c}ois Muzy, 2015. "Hawkes processes in finance," Papers 1502.04592, arXiv.org, revised May 2015.
    2. Peter Halpin & Paul Boeck, 2013. "Modelling Dyadic Interaction with Hawkes Processes," Psychometrika, Springer;The Psychometric Society, vol. 78(4), pages 793-814, October.
    3. 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.
    4. 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.
    5. 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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    Keywords

    Point processes; Hawkes processes; insurance; EM algorithm; natural disasters;
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