Hawkes process

In its seminal paper^[2], Hawkes also considered mutually exciting processes which are now called multivariate Hawkes processes. The ${\textstyle n}$ point processes have arrival times denoted by ${\textstyle 0<t_{1}^{i}<t_{2}^{i}<t_{3}^{i}<\cdots }$ for each type ${\textstyle i=1,\dots ,n}$. The probability of shared points is null, i.e. ${\textstyle \mathbb {P} (\exists (i,k)\neq (j,\ell ):t_{k}^{i}=t_{\ell }^{j})=0}$, and for each ${\textstyle i}$, the infinitesimal probability of an arrival of type ${\textstyle i}$ during the time interval ${\textstyle [t,t+dt)}$ is

${\displaystyle \lambda _{t}^{i}\,dt=\left(\mu _{i}(t)+\sum _{j=1}^{n}\sum _{t_{k}^{j}\,:\,t_{k}^{j}\,<\,t}\phi _{ij}(t-t_{k}^{j})\right)\,dt.}$

The interaction is now described by a matrix of functions ${\textstyle \mathbf {\Phi } =(\phi _{ij})_{1\leq i,j\leq n}}$ and the matrix integral ${\displaystyle \int _{0}^{\infty }\mathbf {\Phi } (t)\,dt}$ plays the role of branching ratio.

Here is an animation to visualize the infinitesimal probabilities ${\displaystyle \lambda _{t}^{i}}$ and associated events ${\displaystyle t_{k}^{i}}$ in a bivariate framework.

Brémaud and Massoulié extended Hawkes processes to allow for a density-dependent birth-rate. They considered an infinitesimal probability of arrival of the following form:

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

for some non-negative function ${\displaystyle f}$.^[3] The original density-independent Hawkes process thus corresponds to ${\displaystyle f(x)=x}$. Of course, both density-dependent and multivariate extensions are possible at the same time. While this density-dependent extension of the Hawkes process is still a point process, it is no longer a branching process because point offspring distributions are no longer independent.

This density-dependent extension allows for negative propensities ${\displaystyle \phi }$. Hence it can model self-inhibition or mutual inhibition (in a multivariate context) which can be important in neuroscience ^[4].

Hawkes processes are used for statistical modeling of events in mathematical finance,^[5] epidemiology,^[6] earthquake seismology,^[7] and other fields in which a random event exhibits self-exciting behavior.^[8][9]

• Point process
• Self-oscillation