Moment generating function of non-Markov self-excited claims processes

This article establishes the moment generating function (mgf) of self-excited claim processes with memory functions that admit a Fourier's transform representation. In this case, the claim and intensity processes may be reformulated as an infinite dimensional Markov processes in the complex plane. Approaching these processes by discretization and next considering the limit allows us to find their moment generating function. We illustrate the article by fitting non-Markov self-excited processes to the time-series of cyber-attacks targeting medical and other services, in the US from 2014 to 2018.