We introduce a class of hybrid marked point processes that encompasses and extends continuous-time Markov chains and Hawkes processes. While this flexible class provides a unified treatment of such existing processes, it also contains new processes with complex dynamics. These marked point processes are defined implicitly via their intensity and are endowed with a state process that interacts with past-dependent events. In fact, this property completely characterises the class of hybrid marked point processes. The key example we entertain is an extension of a Hawkes process, a state-dependent Hawkes process that interacts with its state process. Our main theoretical results establish the existence and uniqueness of hybrid marked point processes under very general assumptions, extending the results of Massouli\'e (1998) on interacting point processes.
We introduce a class of hybrid marked point processes , which encompasses and extends continuous-time Markov chains and Hawkes processes. While this flexible class amalgamates such existing processes, it also contains novel processes with complex dynamics. These processes are defined implicitly via their intensity and are endowed with a state process that interacts with past-dependent events. The key example we entertain is an extension of a Hawkes process, a state-dependent Hawkes process interacting with its state process. We show the existence and uniqueness of hybrid marked point processes under general assumptions, extending the results of Massouli\'e (1998) on interacting point processes.
R
that
, which
R
provides a unified treatment of
amalgamates
R
new
novel
D
marked point
D
In fact, this property completely characterises the class of hybrid marked point processes.
R
that interacts
interacting
R
Our main theoretical results establish
We show
D
very