Hawkes Process

Self-exciting point process: λ(t) = μ + Σ_{t_i < t} α·e^{−β(t−t_i)}

The Hawkes process models self-exciting events (earthquakes, financial trades, social media cascades). Each event raises the conditional intensity λ(t) = μ + Σ α·exp(−β(t−tᵢ)), causing aftershocks. Branching ratio n = α/β: if n < 1, process is subcritical (stable); n ≥ 1 means supercritical (explosive growth). Applications: seismology (Omori's law), high-frequency trading, epidemic modeling.