Markovian multivariate Hawkes population processes: Efficient evaluation of moments

Abstract: We provide probabilistic and computational results on Markovian multivariate Hawkes processes and induced population processes. By applying the Markov property, we characterize in closed form a joint transform, bijective to the probability distribution, of the population process and its underlying intensity process. We demonstrate a method that exploits the transform to obtain analytic expressions for transient and stationary multivariate moments of any order, as well as auto- and cross-covariances. We reveal a nested sequence of block matrices that yields the moments in explicit form and brings important computational advantages. We also establish the asymptotic behavior of the intensity of the multivariate Hawkes process in its nearly unstable regime, under a specific parameterization. In extensive numerical experiments, we analyze the computational complexity, accuracy and efficiency of the established results.