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

Published online by Cambridge University Press:  09 September 2020

Jiwook Jang  and
Rosy Oh Open the ORCID record for Rosy Oh [Opens in a new window]
Jiwook Jang
Affiliation:
Department of Actuarial Studies & Business Analytics, Macquarie Business School, Macquarie University, Sydney, NSW 2109, Australia
Rosy Oh*
Affiliation:
Institute of Mathematical Sciences, Ewha Womans University, Seoul 03760, Korea
*
*Corresponding author. Email: rosy.oh5@gmail.com
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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.

Keywords

Cox processHawkes processShot-noise Poisson processDynamic contagion processCompound process
Type
Review
Information
Annals of Actuarial Science , Volume 15 , Issue 3 , November 2021 , pp. 623 - 644
Copyright
© Institute and Faculty of Actuaries 2020

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