Simulation of linear stationary Hawkes processes: the subexponential case

Linear Hawkes processes form a class of branching point processes that is especially relevant to seismology and epidemiology. This work concerns the simulation of stationary linear Hawkes process when the exact sampling methods do not apply, in the case where the fertility rate of the process properly normalized is the density probability of a subexponential random variable (the known exact sampling method requires a light-tail assumption). This situation is precisely the one arising in the most popular model in seismology (the so-called ETA model, see the main text). Since exact sampling is not available, we have recourse to a simple and natural approximation, the accuracy of which is controlled by a unique parameter. The accuracy of such approximation is measured by the probability that the sample obtained is not typical. We give computable bounds for the accuracy that permit to assess the simulation.