Maximum likelihood identification of Hawkes-Pham models with a guaranteed stability condition

Point processes have many engineering applications and perhaps the most used dynamic system identification model is the Hawkes model. We propose a new approach to maximum likelihood estimation of Hawkes point process models. Although an EM algorithm has previously been given, it turns out to be unreliable in practice. We show that this is because it does not guarantee the stability condition required for the Hawkes process. Our new approach guarantees stability at each iteration. We illustrate with simulations and application to financial data.