Fast and asymptotically efficient estimation in the Hawkes processes

Fast and asymptotically efficient methods for the estimation of the parameters in self-excited counting Hawkes processes are considered. They are based on the Le Cam one-step estimation procedure. An initial guess estimator is given to estimate both the intensity baseline and the parameters of the kernel of the Hawkes process which characterize the influence of an event on the intensity. Then, the estimation is corrected by a single step of a Newton-type gradient-descent algorithm on the loglikelihood function. Asymptotic properties of the one-step estimators are studied. Monte Carlo simulations show the performance of the procedures for finite size samples in terms of computing time and efficiency. The methodology is finally used to study the claim frequency in building insurance.