Hawkes process

In probability theory and statistics, a Hawkes process, named after Alan G. Hawkes, is a kind of self-exciting point process.^[1] It has arrivals at times ${\textstyle 0<t_{1}<t_{2}<t_{3}<\cdots }$ where the infinitesimal probability of an arrival during the time interval ${\textstyle [t,t+dt)}$ is

${\displaystyle \lambda _{t}\,dt=\left(\mu (t)+\sum _{t_{i}\,:\,t_{i}\,<\,t}\phi (t-t_{i})\right)\,dt.}$

The function ${\textstyle \mu }$ is the intensity of an underlying Poisson process. The first arrival occurs at time ${\textstyle t_{1}}$ and immediately after that, the intensity becomes ${\textstyle \mu (t)+\phi (t-t_{1})}$, and at the time ${\textstyle t_{2}}$ of the second arrival the intensity jumps to ${\textstyle \mu (t)+\phi (t-t_{1})+\phi (t-t_{2})}$ and so on.^[2]

During the time interval ${\textstyle (t_{k},t_{k+1})}$, the process is the sum of ${\textstyle k+1}$ independent processes with intensities ${\textstyle \mu (t),\phi (t-t_{1}),\ldots ,\phi (t-t_{k}).}$ The arrivals in the process whose intensity is ${\textstyle \phi (t-t_{k})}$ are the "daughters" of the arrival at time ${\textstyle t_{k}.}$ The integral ${\displaystyle \int _{0}^{\infty }\phi (t)\,dt}$ is the average number of daughters of each arrival and is called the branching ratio. Thus viewing some arrivals as descendants of earlier arrivals, we have a Galton–Watson branching process. The number of such descendants is finite with probability 1 if branching ratio is 1 or less. If the branching ratio is more than 1, then each arrival has positive probability of having infinitely many descendants.