Hawkesprocesser

A Hawkes process (plural: Hawkes processes, sometimes written Hawkesprocesser) is a class of stochastic point processes characterized by self-excitation: the occurrence of an event increases the probability of future events for some period. Introduced by Alan G. Hawkes in 1971, these processes are defined through a conditional intensity function λ(t) that depends on a baseline rate and a sum of kernel contributions from past events, typically written λ(t) = μ(t) + Σ φ(t − ti) for event times ti < t. The kernel φ determines how past events influence future intensity; common choices are exponential or power-law kernels.

Hawkes processes admit a branching representation, interpreting events as immigrants (from the baseline) and their offspring (triggered events). This structure yields conditions for stationarity and stability—most notably that the branching ratio (total expected offspring per event) must be less than one. Variants include multivariate (mutually-exciting) processes, marked processes, and nonlinear Hawkes models.

They are used in diverse fields where clustered temporal events occur: seismology (aftershock models such as ETAS), finance (modeling trades and price moves), social media (cascade dynamics), neuroscience (spike trains), crime and epidemiology. Parameter estimation commonly uses maximum likelihood, EM algorithms, or nonparametric kernel estimation; computation can be intensive for large datasets. Limitations include model misspecification, identifiability challenges, and sensitivity to kernel choice, motivating ongoing research into scalable inference and flexible kernel learning.