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A review on Poisson, Cox, Hawkes, shot-noise Poisson and dynamic contagion process and their compound processes
Annals of Actuarial Science Pub Date : 2020-09-09 , DOI: 10.1017/s1748499520000287
Jiwook Jang , Rosy Oh

The Poisson process is an essential building block to move up to complicated counting processes, such as the Cox (“doubly stochastic Poisson”) process, the Hawkes (“self-exciting”) process, exponentially decaying shot-noise Poisson (simply “shot-noise Poisson”) process and the dynamic contagion process. The Cox process provides flexibility by letting the intensity not only depending on time but also allowing it to be a stochastic process. The Hawkes process has self-exciting property and clustering effects. Shot-noise Poisson process is an extension of the Poisson process, where it is capable of displaying the frequency, magnitude and time period needed to determine the effect of points. The dynamic contagion process is a point process, where its intensity generalises the Hawkes process and Cox process with exponentially decaying shot-noise intensity. To facilitate the usage of these processes in practice, we revisit the distributional properties of the Poisson, Cox, Hawkes, shot-noise Poisson and dynamic contagion process and their compound processes. We provide simulation algorithms for these processes, which would be useful to statistical analysis, further business applications and research. As an application of the compound processes, numerical comparisons of value-at-risk and tail conditional expectation are made.

中文翻译:

泊松、考克斯、霍克斯、散粒噪声泊松和动态传染过程及其复合过程综述

泊松过程是升级到复杂计数过程的重要组成部分,例如 Cox(“双随机泊松”)过程、霍克斯(“自激”)过程、指数衰减散粒噪声泊松(简称“散粒-noise Poisson”)过程和动态传染过程。Cox 过程通过让强度不仅取决于时间而且还允许它是一个随机过程来提供灵活性。霍克斯过程具有自激特性和聚类效应。散粒噪声泊松过程是泊松过程的扩展,它能够显示确定点效应所需的频率、幅度和时间段。动态传染过程是一个点过程,它的强度概括了霍克斯过程和考克斯过程,散粒噪声强度呈指数衰减。为了便于在实践中使用这些过程,我们重新审视了泊松、考克斯、霍克斯、散粒噪声泊松和动态传染过程及其复合过程的分布特性。我们为这些过程提供模拟算法,这将有助于统计分析、进一步的业务应用和研究。作为复合过程的一种应用,对风险价值和尾部条件期望进行了数值比较。这将有助于统计分析、进一步的商业应用和研究。作为复合过程的一种应用,对风险价值和尾部条件期望进行了数值比较。这将有助于统计分析、进一步的商业应用和研究。作为复合过程的一种应用,对风险价值和尾部条件期望进行了数值比较。
更新日期:2020-09-09
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