  This work contributes to the theory and applications of Hawkes processes. We
introduce and examine a new class of Hawkes processes that we call generalized
Hawkes processes, and their special subclass -- the generalized multivariate
Hawkes processes (GMHPs). GMHPs are multivariate marked point processes that
add an important feature to the family of the (classical) multivariate Hawkes
processes: they allow for explicit modelling of simultaneous occurrence of
excitation events coming from different sources, i.e. caused by different
coordinates of the multivariate process. We study the issue of existence of a
generalized Hawkes process, and we provide a construction of a specific
generalized multivariate Hawkes process. We investigate Markovian aspects of
GMHPs, and we indicate some plausible important applications of GMHPs.
