Quick breakdown of why Hawkes matters here:
A standard Poisson process (used in classic Merton) has no memory. The probability of the next jump is the same whether a jump just happened or not.
A Hawkes process is self-exciting — each arriving event temporarily raises the rate of future events. The excitation decays exponentially: λ(t) = λ₀ + α · Σ exp(−β · (t − tᵢ))
The key constraint: α/β < 1 keeps the process stationary. Push past that and intensity explodes.
In practice, this means a single bad print can cascade — and the simulation captures exactly that.