Hybrid marked point processes: characterisation, existence and uniqueness

TL;DR

This paper introduces a new class of hybrid marked point processes that generalize and extend existing models like Markov chains and Hawkes processes, allowing for complex, state-dependent dynamics with proven existence and uniqueness.

Contribution

It defines hybrid marked point processes, including novel state-dependent models, and establishes their existence and uniqueness under broad conditions.

Findings

Hybrid processes encompass and extend Markov chains and Hawkes processes.

Existence and uniqueness are proven under general assumptions.

Includes a state-dependent Hawkes process with interacting dynamics.

Abstract

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.