Published online by Cambridge University Press: 18 September 2020
A very interesting class of stochastic processes was introduced by Alan Hawkes (1971). These processes, now called Hawkes processes, are meant to model self-exciting and mutually-exciting random phenomena that evolve in time. The self-exciting phenomena are modeled as univariate Hawkes processes, and the mutually-exciting phenomena are modeled as multivariate Hawkes processes. Hawkes processes belong to the family of marked point processes, and, of course, a univariate Hawkes process is just a special case of the multivariate one. In this chapter we define and study generalized multivariate Hawkes processes, as well as the related consistencies and structures. Generalized multivariate Hawkes processes are multivariate marked point processes that add an important feature to the family of (classical) multivariate Hawkes processes: they allow for explicit modeling of simultaneous occurrence of excitation events coming from different sources, i.e. caused by different coordinates of the multivariate process.