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Title: A Hawkes model with CARMA(p,q) intensity Authors:  Lorenzo Mercuri - University of Milan (Italy) [presenting]
Andrea Perchiazzo - Vrije Universiteit Brussel (Belgium)
Edit Rroji - Universita' degli studi di Milano-Bicocca (Italy)
Abstract: A new model named CARMA$(p,q)$-Hawkes process is introduced. The Hawkes model with exponential kernel implies a strictly decreasing behaviour of the autocorrelation function, and empirical evidence rejects 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.