We study the Hawkes process and its main components: conditional intensity, kernel function and branching ratio, deriving the conditions for asymptotic stationarity, extinction and explosion. The interpretation of Hawkes processes as cluster processes or immigration-birth cluster processes is explained. We present some results about the number and lenght of clusters, using the concept of the probability generating functional. By using the compensator and the random time change theorem, a Hawkes process can be transformed into a homogeneous Poisson process, which can be used for goodness-of-fit test. The various algorithms for generating Hawkes process are presented: the thinning algorithm by Ogata, the clustering algorithm and perfect simulation of weakly stationary Hawkes process. The multivariate Hawkes processes are introduced and its corresponding simulation algorithm is implemented. In conclusion, we present some applications of Hawkes processes in research.
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