We consider Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of the Hawkes process impacts the intensity of the random point process by the addition of a signed reproduction function. The case of a non-negative reproduction function corresponds to self-excitation; it has been largely investigated in the literature and is well understood. In particular, there then exists a cluster representation of the self-excited Hawkes processes which allows to apply results known for continuous-time age-structured Galton-Watson trees to these random point processes. In the case we study, the cluster representation is no longer valid, and we use renewal techniques. We establish limit results for Hawkes process with signed reproduction functions, notably generalizing exponential concentration inequalities proved by Reynaud-Bouret and Roy (2007) for non-negative reproduction functions. An important step is to establish the existence of exponential moments for the distribution of renewal times of M/G/1 queues that appear naturally in our problem.
This is a joint work with M. Costa C. Graham and L. Marsalle.