Hawkes process

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.

• 1 Applications
• 2 See also
• 3 References
• 4 Further reading

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