  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.
