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.

霍克斯过程是时间自激点过程。他们在地震建模或金融领域建立了良好的基础，并且他们的应用正在扩展到不同的领域。文献中的大多数模型在其潜在的保险应用方面有两个主要缺点。首先，他们使用了一种指数衰减的激励形式，这不允许事件的发生与其对过程的激励影响之间存在延迟，因此不适用于保险数据。其次，从这些模型中得出的理论结果仅在观察时间趋于无穷大时才有效，而保险用例的时间范围是几个月或几年。在本文中，我们定义了一个完整的具有伽马密度激励函数的霍克斯过程框架（即估计、模拟、拟合优度）而不是指数衰减函数，我们展示了关于过程瞬态状态的一些数学特性（即期望、方差）。我们用卢森堡自然灾害的真实保险数据来说明我们的结果。