TY - JOUR
T1 - A review on Poisson, Cox, Hawkes, shot-noise Poisson and dynamic contagion process and their compound processes
AU - Jang, Jiwook
AU - Oh, Rosy
PY - 2020/9/9
Y1 - 2020/9/9
N2 - 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.
AB - 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.
KW - Compound process
KW - Cox process
KW - Dynamic contagion process
KW - Hawkes process
KW - Shot-noise Poisson process
UR - http://www.scopus.com/inward/record.url?scp=85092314743&partnerID=8YFLogxK
U2 - 10.1017/S1748499520000287
DO - 10.1017/S1748499520000287
M3 - Review article
AN - SCOPUS:85092314743
JO - Annals of Actuarial Science
JF - Annals of Actuarial Science
SN - 1748-4995
ER -