Hawkes autoregressive processes: a new model for multiscale and heterogeneous processes

TL;DR

This paper introduces the Hawkes autoregressive (HAR) model that combines continuous- and discrete-time dynamics to better represent multiscale, heterogeneous data, with theoretical guarantees on its properties.

Contribution

The paper proposes a novel HAR model integrating Hawkes and autoregressive processes, and provides foundational probabilistic results including stationarity and stability.

Findings

Established existence of a stationary version of HAR

Derived a cluster representation for the model

Proved stability and ergodic properties of HAR

Abstract

Both Hawkes processes and autoregressive processes rely on linear functionals of their past, while modeling different types of data. Since datasets arising from observations of the same phenomenon may be heterogeneous and sampled at different time scales, it is natural to study multiscale and heterogeneous processes, such as those obtained by combining Hawkes and autoregressive dynamics. In this paper, we introduce this new Hawkes autoregressive (HAR) model incorporating both continuous- and discrete-time dynamics, and establish several probabilistic results, including the existence of a stationary version, a cluster representation, as well as stability and ergodic properties.