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

A Hawkes model with CARMA(p,q) intensity