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

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.

Drug Treatment Effect Model Based on MODWT and Hawkes Self-Exciting Point Process

“…The essential property of self-exciting point process is the occurrence of any event increased the probability of further events occurring which is consistent with the behavior of drug on patient. This model origins from the area of earthquake behavior analysis [25] and is recently extended to the fields of finance [26], disease prediction [27], and social media [28], where the model has shown satisfactory performance, but it has not been used for drug effect modelling yet.…”