Hawkes process

Short description: Self-exciting counting process

In probability theory and statistics, a Hawkes process is an age-dependent branching process driven by immigration from an inhomogeneous Poisson process. The process, named after Alan G. Hawkes, is also called a 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

${\lambda }_{t}dt=(\mu (t)+\sum _{t_k:t_k<t}\varphi (t-t_k))dt.$

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 $\mu (t)+\varphi (t-t_1)$, and at the time $t_2$ of the second arrival the intensity jumps to $\mu (t)+\varphi (t-t_1)+\varphi (t-t_2)$ and so on.

During the time interval $(t_k,t_{k+1})$, the process is the sum of $k+1$ independent processes with intensities $\mu (t),\varphi (t-t_1),\dots ,\varphi (t-t_k).$ The arrivals in the process whose intensity is $\varphi (t-t_k)$ are the "daughters" of the arrival at time $t_k.$ The integral $\underset{0}{\overset{\mathrm{\infty }}{\int }}\varphi (t)dt$ 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.

Multivariate extension

In its seminal paper, Hawkes also considered mutually exciting processes which are now called multivariate Hawkes processes. The $n$ point processes have arrival times denoted by $0<t_1^i<t_2^i<t_3^i<\cdots$ for each type $i=1,\dots ,n$. The probability of shared points is null, i.e. $\mathbb{\mathbb{P}}(\mathrm{\exists }(i,k)\ne (j,\ell ):t_k^i=t_{\ell }^j)=0$, and for each $i$, the infinitesimal probability of an arrival of type $i$ during the time interval $[t,t+dt)$ is

${\lambda }_t^{i}dt=({\mu }_i(t)+\sum _{j=1}^n\sum _{t_k^j:t_k^j<t}{\varphi }_{ij}(t-t_k^j))dt.$

The interaction is now described by a matrix of functions $\mathrm{\Phi }=({\varphi }_{ij}{)}_{1\le i,j\le n}$ and the matrix integral $\underset{0}{\overset{\mathrm{\infty }}{\int }}\mathrm{\Phi }(t)dt$ plays the role of branching ratio.

Here is an animation to visualize the infinitesimal probabilities ${\lambda }_t^i$ and associated events $t_k^i$ in a bivariate framework.

Density-dependent extension

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:

${\lambda }_{t}dt=f(\mu (t)+\sum _{t_k:t_k<t}\varphi (t-t_k))dt,$

for some non-negative function $f$. The original density-independent Hawkes process thus corresponds to $f(x)=x$. 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 $\varphi$. Hence it can model self-inhibition or mutual inhibition (in a multivariate context) which can be important in neuroscience .

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.

See also

• Point process
• Self-oscillation

Further reading

• Bacry, Emmanuel; Mastromatteo, Iacopo; Muzy, Jean-François (2015). "Hawkes processes in finance". arXiv:1502.04592 [q-fin.TR].
• Rizoiu, Marian-Andrei; Lee, Young; Mishra, Swapnil; Xie, Lexing (2017). "A Tutorial on Hawkes Processes for Events in Social Media". arXiv:1708.06401 [stat.ML].