Hawkes Process

The Hawkes process, introduced by Alan Hawkes in 1971, models sequences of events where each event temporarily increases the probability of subsequent events. Unlike a Poisson process (where events occur independently), a Hawkes process is "self-exciting" — a financial shock at time t₀ elevates the probability of further shocks for a period determined by the decay function. The intensity (instantaneous event rate) at any time is: λ(t) = μ + Σ α·φ(t - tᵢ), where μ is the baseline rate, α is the excitation parameter, and φ is the decay kernel.

The critical parameter is the branching ratio n = α/β (where β is the decay rate). When n < 1, the process is subcritical — each shock cluster eventually dies out. When n > 1, the process is supercritical — shocks self-sustain and amplify indefinitely. The transition from n = 0.9 to n = 1.1 represents the phase transition between "volatile but stable" and "self-sustaining crisis cascade". This is directly analogous to the R₀ concept in epidemiology — when the financial contagion reproduction number exceeds 1, the crisis spreads without bound.

In sovereign risk analysis, the Hawkes process is calibrated using historical default events, CDS spread spikes, and bank equity crashes within each country. A branching ratio above 0.8 signals the system is approaching the supercritical threshold — one additional shock could trigger a cascade. Above 1.0, the HAWKES_SUPERCRITICAL alert is activated in the SIGMA Engine, indicating the current event sequence is self-reinforcing rather than self-limiting.