Stability and perfect simulation of non-linear Hawkes Processes with Erlang kernels

We consider non-linear Hawkes processes with Erlang memory kernels and show that we can relate their stability properties to the study of an associated system of Piecewise Deterministic Markov processes (PDMPu0027s). We state explicit conditions implying the positive Harris recurrence of the process. The proof is based on integration by parts with respect to the jump times. A crucial property is the non-degeneracy of the transition semigroup which is obtained thanks to the invertibility of an associated Vandermonde matrix. In the case of Lipschitz continuous rate functions we also provide explicit bounds for the exponential rate of convergence of the process to equilibrium in Wasserstein distance. As a consequence, we are able to provide a numerically efficient simulation algorithm for non-linear Hawkes process with Erlang kernels. Our method is able to exactly generate the point process and intensity process. Moreover, it is flexible to generate points with either stationary or non-stationary intensity, starting from any arbitrary time with any arbitrary initial intensity. The method is straightforward to implement, and can be easily extended to multi-dimensional versions.