A Hawkes model with CARMA(p,q) intensity

TL;DR

This paper introduces the CARMA(p,q)-Hawkes process, a new model that captures more realistic dependence structures in event data by combining Hawkes processes with CARMA intensity dynamics, addressing limitations of exponential kernels.

Contribution

The paper proposes the CARMA(p,q)-Hawkes process, extending Hawkes models with CARMA-based intensities to better model complex autocorrelation structures.

Findings

The model can reproduce non-monotonic autocorrelation functions.

Conditions for stationarity and positivity are established.

Simulation and estimation methods are developed and demonstrated.

Abstract

In this paper we introduce a new model named CARMA(p,q)-Hawkes process as the Hawkes model with exponential kernel implies a strictly decreasing behaviour of the autocorrelation function and empirically evidences reject the monotonicity assumption on the autocorrelation function. The proposed model is a Hawkes process where the intensity follows a Continuous Time Autoregressive Moving Average (CARMA) process and specifically is able to reproduce more realistic dependence structures. We also study the conditions of stationarity and positivity for the intensity and the strong mixing property for the increments. Furthermore we compute the likelihood, present a simulation method and discuss an estimation method based on the autocorrelation function. A simulation and estimation exercise highlights the main features of the CARMA(p,q)-Hawkes.