Hawkes Process

The term "Hawkes process" encompasses a number of point process notions; while many are similar, some are rather different. There are also different formulations for univariate and multivariate point processes.

In some literature, a univariate Hawkes process is defined to be a self-exciting temporal point process whose conditional intensity function is defined to be

(1)

where is the background rate of the process , where are the points in time occurring prior to time , and where is a function which governs the clustering density of . The function is sometimes called the exciting function or the excitation function of . Similarly, some authors (Mehrdad and Zhu 2014) denote the conditional intensity function by and rewrite the addend in (1) as

(2)

The processes upon which Hawkes himself made the most progress were univariate self-exciting temporal point processes whose conditional intensity function depends linearly on the past arrivals (Hawkes 1971). As a result, some authors refer to such processes as Hawkes processes. In general, however, such behavior is typically specified, i.e., processes for which is linear are referred to as linear Hawkes processes and are differentiated from their nonlinear counterparts whose conditional intensity functions are nonlinear (Mehrdad and Zhu 2014).

Still other authors consider two alternative brands of univariate Hawkes processes, one the so-called intensity-based Hawkes process and the other the so-called cluster-based version, which are equivalent though are studied in different contexts (Dassios and Zhao 2013). In this case, the intensity-based process is a temporal point process on which has a nonnegative exponentially-decaying -stochastic intensity of the form

(3)

for , where is a history of the process with respect to which is adapted, is the constant reversion level, is the initial intensity at time , is the constant rate of exponential decay, are the sizes of the self-excited jumps viewed as independent random variables distributed according to some distribution function , , and each new mark is independent of its arrival time and the preceding history. The mark affects subsequent arrival times, so the full sequences and are generally dependent. This is equivalent to the cluster-based version in which (3) is viewed as a marked Poisson cluster process , the only difference being that from the cluster-based perspective:

1. The set consists of elements known as immigrants which are distributed as an inhomogeneous Poisson process with rate

(4)

for .

2. The set of marks associated to the immigrants are independent of the immigrants and are distributed as independent random variables according to some distribution .

3. Each immigrant generates a single cluster independent of other clusters where here, each is viewed as a random set subject to a certain branching structure (Dassios and Zhao 2013) which satisfies the property that .

In addition to these ambiguities, several authors (e.g., Mehrdad and Zhu 2014), have made generalized versions of the univariate process described in (1) which are still referred to as Hawkes processes. One example includes adapting (1) so that the process has different exciting functions, the result of which is a collection of non-explosive simple point processes for which:

1. is an inhomogeneous Poisson process with intensity at time ;

2. For every , is a simple point process with intensity

(5)

3. For every , is an inhomogeneous Poisson process with intensity conditional on .

In this context, provided that the superposition is non-explosive, meaning that it has finitely many points in every bounded time interval almost surely, is said to be a univariate Hawkes point process with excitation functions while is called the immigrant process and the th generation offspring process (Mehrdad and Zhu 2014). Note that when and when for any , this extended model reduces to the classical linear model (2).

Due to the vast usage of the term Hawkes process among univariate point processes, one expects there to be room for an equally large number of definitions of multivariate Hawkes processes. Surprisingly, however, the most common use of the term is assigned to a relatively straightforward extension of equations (1) and (2), whereby one says that a multivariate -dimensional counting process taking values in is a multivariate Hawkes process whenever the associated conditional intensity function defined by

(6)

for , ..., has the form

(7)

(Bacry et al. 2012). Here, stands for probability, is the sigma-algebra generated by up to present time , , and for .

It is worth noting, however, that precisely as with the univariate case, some authors distinguish between different "types" of multivariate Hawkes processes (Liniger 2009) while other authors define completely separate types of multivariate functions to be multivariate Hawkes processes (Carlsson et al. 2007).