Hawkes Processes: Self-Exciting Point Processes

• Hawkes processes are temporal point processes defined by their self-exciting property, where each event increases the likelihood of future events.
• They utilize a branching structure that models clusters of events, making them valuable in finance, neuroscience, seismology, and epidemiology.
• Generalizations include multivariate extensions, nonlinear link functions, and deep-learning-based intensities to capture complex interactions.

Hawkes processes are a class of temporal point processes characterized by the self-exciting property, wherein the occurrence of an event increases the conditional intensity, and thus the likelihood, of subsequent events over some time scale. The stochastic intensity function endows these processes with a branching structure, leading to clustering effects, long-range dependence, and tractable statistical inference in both univariate and high-dimensional multivariate settings. Originally introduced to model earthquake aftershocks, Hawkes processes now underpin quantitative frameworks across finance, neuroscience, social sciences, seismology, epidemiology, and engineered systems. The field has seen extensive generalizations including nonlinear link functions, marks, graph-structured interactions, stochastic excitation levels, variable memory forms, renewal immigrant arrivals, and deep-learning-based intensities.

1. Mathematical Structure and Fundamental Properties

A univariate Hawkes process is a simple counting process $N(t)$ on $[0,T]$ with conditional intensity

$\lambda(t) = \mu + \int_0^t \varphi(t-s) \, dN(s),$

where $\mu > 0$ is the exogenous baseline rate, and $\varphi : (0,\infty) \to [0,\infty)$ is the excitation kernel, typically normalized so that $\eta = \int_0^\infty \varphi(s) ds < 1$ to guarantee subcritical (stationary) dynamics (Laub et al., 2015, Laub et al., 2024). Each event increases the intensity by $\varphi(t-\tau_i)$ and the process clusters due to recursively generated “offspring” events.

Generalizations include:

• Multivariate Hawkes: For $\bm{N}(t) = (N^1(t), ..., N^d(t))^\top$,

$$\lambda_i(t) = \mu_i + \sum_{j=1}^d \int_0^t \varphi_{ij}(t-s) \, dN^j(s),$$

with $\varphi_{ij}$ measuring the mutual or self-excitatory influence from $[0,T]$0 to $[0,T]$1. The process is stationary if the spectral radius $[0,T]$2 for $[0,T]$3 (Laub et al., 2015, Bacry et al., 2015).

• Branching Representation: Hawkes processes are Poisson cluster processes: immigrants arrive with rate $[0,T]$4, each spawning a branching tree of offspring governed by $[0,T]$5. The branching ratio $[0,T]$6 is the expected number of first-generation offspring. Stationarity, ergodicity, and the law of large numbers $[0,T]$7 follow for $[0,T]$8 (Laub et al., 2024, Laub et al., 2015).
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