The Hawkes process is a model for counting the number of arrivals to a system which exhibits the self-exciting property - that one arrival creates a heightened chance of further arrivals in the near future. The model, and its generalizations, have been applied in a plethora of disparate domains, though two particularly developed applications are in seismology and in finance. As the original model is elegantly simple, generalizations have been proposed which: track marks for each arrival, are multivariate, have a spatial component, are driven by renewal processes, treat time as discrete, and so on. This paper creates a cohesive review of the traditional Hawkes model and the modern generalizations, providing details on their construction, simulation algorithms, and giving key references to the appropriate literature for a detailed treatment.

翻译：霍克斯过程是一种用于统计系统到达次数的模型，具有自激特性——即一次到达会在近期内引发更高概率的后续到达。该模型及其推广形式已广泛应用于众多不同领域，其中尤以地震学和金融学领域的发展最为成熟。由于原始模型设计简洁优雅，后续学者提出了包含以下特征的推广版本：为每次到达标记特征、多变量模型、空间分量、基于更新过程驱动、离散时间处理等。本文对传统霍克斯模型及其现代推广进行了系统性综述，详细阐述了其构建方法、模拟算法，并提供了关键参考文献以便读者进行深入研读。