Hawkes processes: when past events trigger future ones
Add a kick to the intensity every time an event fires, and let it decay back. The result is a point process with realistic clustering — trade arrivals, order-book events, default cascades all look like this.
A Poisson process arrives at events independently of its past at a constant rate $\mu$. A Hawkes process (Hawkes 1971) adds *self-excitation*: every event bumps the arrival intensity upward by $\alpha$, then the intensity decays back to baseline at rate $\beta$. The result is a point process with built-in clustering — quiet periods, then bursts, then quiet again — that fits trade-arrival, order-book-event, and default-cascade data far better than Poisson. The model has three parameters ($\mu, \alpha, \beta$) and one critical quantity: the branching ratio $\eta = \alpha/\beta$. Below $\eta = 1$ the process is stable; above $\eta = 1$ it explodes; near $\eta = 1$ it scales to rough vol (Jaisson-Rosenbaum 2015), connecting microstructure directly to rough-vol modelling.
β Intro Β· expand
Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine β being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
60s
β Try first Β· expand
Worked example
A Hawkes process models trade arrivals on a quiet equity ticker with baseline intensity $\mu = 0.5$ trades per second, jump $\alpha = 1.0$ per event, and decay rate $\beta = 2.0$ per second. (a) Compute the branching ratio. (b) Mean cluster size triggered by one exogenous event. (c) Long-run (stationary) intensity.
β Worked example Β· expand
Practice 1 of 3Type a fraction, decimal, or expression β mathjs parses it.
β Practice Β· expand
Reflection
Hawkes and Galton-Watson both have a criticality threshold ($\eta = 1$ and $\mu = 1$ respectively). What is structurally different about the two processes that makes one a continuous-time intensity model and the other a discrete-time genealogy?