A Poisson process has a constant arrival rate — events don't care about the past. Markets
aren't like that: a trade makes the next trade more likely, an order triggers a cascade.
A Hawkes process captures this with an intensity that jumps up at every event and decays
between them. That is the core idea in Ibrahim Lanre Adedimeji’s “Hawkes Processes: A
Stochastic Gem.”
The self-exciting intensity
The conditional intensity (instantaneous arrival rate given the history) is
λ(t)=μ+ti<t
References and further study
Know what to read—and why
Optional further reading only — Wikipedia, OpenStax, author sites, and MIT OCW course pages. Prefer open encyclopedia / CC BY texts first. These links are not MIT endorsement, and MatheLinux lessons are not hosted copies of OCW.
Copyright rule: MatheLinux stores only the citation. It does not copy or host external books, papers, videos, or datasets. MIT OCW is CC BY-NC-SA (non-commercial); open it on ocw.mit.edu. Publisher books require a legal copy. Quote sparingly and cite.
Submit work
Before you continue
Lesson close · Hawkes Processes & Market Microstructure
ϕ(⋅)≥0 — the excitation kernel: how much each past event lifts the rate.
The standard choice is an exponential kernel:
ϕ(t)=αe−βt,α,β>0.
Every event adds α to the intensity, which then decays at rate β. So bursts beget
bursts — and then quiet returns. Power-law kernels are also used when memory is longer; the
exponential case is the clean Markov embedding for control and simulation.
Stability and the branching ratio
Each event spawns, on average,
n=βα=∫0∞ϕ(t)dt
"offspring" events. This branching ratio must satisfy n<1 for the process to be
stable (otherwise activity explodes). Read n as the fraction of activity that is
endogenous — self-generated by the market rather than driven by outside news. Empirically
n is high (often 0.7–0.95) in liquid markets: most activity is the market reacting to
itself.
When n sits near 1, small news can trigger long cascades — the same qualitative picture
as critical branching in other self-exciting systems.
Why Poisson fails on the tape
Under Poisson arrivals, inter-event times are memoryless and variance equals mean. Real
order flow is overdispersed and clustered: trades arrive in bursts, then go quiet. Hawkes
recovers that clustering without inventing a separate “regime” variable — the history is
the regime, stored in the intensity.
Why quants care
Hawkes processes reproduce volatility clustering and trade clustering that Poisson
models can't. The next lessons plug this intensity into a jump-diffusion price and into the
order book — the path from Ibrahim’s “Capturing Volatility Clustering Using Hawkes-Driven
Jump Diffusion Models” into multivariate LOB dynamics.
Free preview of the Hawkes & Microstructure course.
Based on Ibrahim Lanre Adedimeji’s “Hawkes Processes: A Stochastic Gem.”
Interactive · Hawkes intensity / branching ratio
branching n=α/β=0.667 (stable) · events≈0
Mark this lesson complete to track progress
Knowledge check
Q1.For stability of a Hawkes process with kernel αe^(−βt), the branching ratio n = α/β must satisfy…
Q2.A branching ratio measured at n ≈ 0.9 tells you that market activity is mostly…
Submit work
Choose one lesson task
Optional human feedback—not required to mark the lesson complete. Pick a brief, submit it, or continue without submitting.
Optional
Human-graded work is a Pro course feature
The free lesson, quiz, and problem checks remain available. Pro members can choose a brief below, submit work, and receive a score and written feedback.
