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Hawkes Process

A self-exciting point process in which every event temporarily raises the probability of further events occurring soon afterward, so that occurrences cluster in bursts rather than scattering independently, the standard model for earthquake aftershocks, contagious disease spread, and order-flow clustering in financial markets.

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In Plain English

In a plain Poisson process, one event has no effect on the odds of the next, events are completely independent. A Hawkes process breaks that independence deliberately: every event gives a temporary boost to the intensity (the instantaneous event rate), making further events more likely right afterward, and that boost fades away over time. This produces exactly the kind of bursty, clustered pattern real event data often shows, an earthquake triggers aftershocks, one large trade in a market triggers a flurry of related trades, one viral social media post triggers a wave of related posts, all far more clustered in time than pure randomness would produce.

Definition

A Hawkes process is a point process with conditional intensity $λ (t) = μ + \sum_{t_{i} < t} ϕ (t - t_{i})$ , where $μ > 0$ is the baseline (background) intensity, and $ϕ (\cdot) \geq 0$ is the excitation kernel, a function specifying how much and for how long each past event at time $t_{i}$ raises the intensity at the current time $t$ . The most common kernel choice is exponential decay, $ϕ (s) = α e^{- β s}$ , giving each event an initial intensity boost of $α$ that decays at rate $β$ . The branching ratio $n = \int_{0}^{\infty} ϕ (s) d s$ (equal to $α / β$ for the exponential kernel) determines stability: the process is stationary and well-behaved when $n < 1$ (each event triggers, on average, fewer than one further 'child' event), and explosive when $n \geq 1$ . Because past events feed back into current intensity, a Hawkes process is not a Poisson process and not Markov in event counts alone (though it can be represented as Markov by augmenting the state with the current intensity level). It underlies the widely used ETAS (Epidemic-Type Aftershock Sequence) model of seismology and is a standard tool in high-frequency financial econometrics for modeling order-arrival clustering.

Formula

λ (t) = μ + t_{i} < t \sum ϕ (t - t_{i}), ϕ (s) = α e^{- β s}

Notation

λ (t)

—Conditional intensity: instantaneous event rate given the history up to time $t$

μ

—Baseline (background) intensity, independent of past events

ϕ (s)

—Excitation kernel: intensity boost contributed by a past event, $s$ time units ago

n

—Branching ratio, $\int_0^\infty \phi(s)\,ds$; stability requires $n < 1$

Properties

Named after Alan Hawkes, who introduced the model in 1971, the Hawkes process is now the standard, most widely used mathematical formalization of the broader self-exciting process idea, most applied work that says 'self-exciting model' means, specifically, some variant of the Hawkes process.
The branching ratio $n$ has a clean interpretation as the expected number of 'child' events directly triggered by a single 'parent' event, this branching-process reinterpretation (each event spawning a random number of further events) is exactly why Hawkes processes connect naturally to the theory of branching processes and to epidemic reproduction-number intuition.
Because the intensity depends on the entire event history (not just the current state), Hawkes processes are not Markov in the raw event-count sense, but for exponential kernels the intensity itself can be tracked as a Markov process, an important trick that makes both simulation and likelihood-based estimation computationally tractable.

At a Glance

Intensity structurebaseline rate plus a decaying contribution from every past event

Common kernelexponential decay,

ϕ (s) = α e^{- β s}

Stability conditionbranching ratio

n < 1

for a stationary, non-explosive process

Signature applicationsearthquake aftershocks (ETAS model), order-flow clustering in finance, epidemic contagion

Last updated September 8, 2026← Back to the Dictionary