Hawkes process

In probability theory and statistics, a Hawkes process, named after Alan G. Hawkes, is a kind of self-exciting point process. It has arrivals at times $ 0 < t_1 < t_2 < t_3 < \cdots $ where the infinitesimal probability of an arrival during the time interval $ [t,t+dt) $  is


 * $$ \lambda_t \, dt = \left( \mu(t) + \sum_{t_i\,:\, t_i\,<\,t} \phi(t-t_i) \right) \, dt. $$

The function $\mu$ is the intensity of an underlying Poisson process. The first arrival occurs at time $ t_1$ and immediately after that, the intensity becomes $ \mu(t) + \phi(t-t_1) $, and at the time $ t_2$  of the second arrival the intensity jumps to $ \mu(t) + \phi(t-t_1) + \phi(t-t_2) $  and so on.

During the time interval $ (t_k, t_{k+1}) $, the process is the sum of $ k+1$ independent processes with intensities $ \mu(t), \phi(t-t_1), \ldots, \phi(t-t_k). $ The arrivals in the process whose intensity is $ \phi(t-t_k) $  are the "daughters" of the arrival at time $ t_k.$  The integral $$ \int_0^\infty \phi(t)\,dt$$ 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.

Applications
Hawkes processes are used for statistical modeling of events in mathematical finance, epidemiology, earthquake seismology, and other fields in which a random event exhibits self-exciting behavior.