A review on Poisson, Cox, Hawkes, shot-noise Poisson and dynamic contagion process and their compound processes

Jiwook Jang, Rosy Oh*

*Corresponding author for this work

Research output: Contribution to journalReview articlepeer-review

Abstract

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.

Original languageEnglish
JournalAnnals of Actuarial Science
Early online date9 Sep 2020
DOIs
Publication statusE-pub ahead of print - 9 Sep 2020

Keywords

  • Compound process
  • Cox process
  • Dynamic contagion process
  • Hawkes process
  • Shot-noise Poisson process

Fingerprint Dive into the research topics of 'A review on Poisson, Cox, Hawkes, shot-noise Poisson and dynamic contagion process and their compound processes'. Together they form a unique fingerprint.

Cite this