Simulation , Estimation and Applications of Hawkes Processes

Hawkes processes are a particularly interesting class of stochastic processes that were introduced in the early seventies by A. G. Hawkes, notably to model the occurrence of seismic events. Since then they have been applied in diverse areas, from earthquake modeling to financial analysis. The processes themselves are characterized by a stochastic intensity vector, which represents the conditional probability density of the occurrence of an event in the immediate future. They are point processes whose defining characteristic is that they self-excite, meaning that each arrival increases the rate of future arrivals for some period of time. In this project, we present background and all major aspects of Hawkes processes. Before introducing the univariate Hawkes process, we recall the theory regarding counting processes and Poisson processes. Then we provide two equivalent definitions of Hawkes process and the theory regarding stochastic time change, which is necessary in the analysis of Hawkes processes. We show two simulation algorithms, using two different approaches, which let us generate Hawkes processes. Then we define what a multivariate Hawkes process is. We also present a real data example, which is the analysis of the discharge of one antennal lobe neuron of a cockroach during a spontaneous activity. We provide a plot of this data and we estimate the intensity process, using the methods developed by Christophe Pouzat. Finally, we discuss possible applications of Hawkes processes as well as possibilities for future research in the area of self-exciting processes.