Hawkes Processes with Variable Base Intensity Driven by a Markov Chain

Hawkes processes are a class of point processes renowned for their ability to model self-exciting phenomena, wherein an event’s occurrence significantly influ- ences the future behaviour of the process, thereby increasing the likelihood of sub- sequent events. Traditionally, these models have relied on assumptions regarding the deterministic or constant nature of process parameters. This poses a challenge in parameter estimation, typically addressed through the maximization of the log- likelihood function. This study aims to address these limitations by considering the base intensity parameter of the Hawkes process as a continious-time Markov chain. Our main interest in this work is to study the consistency and asymptotic normality of parameter estimates for Hawkes processes with variable base intensity using Hamilton filter. The performance of the approach is assessed using simulation experiments and compared with classical log-likelihood estimation for point pro- cesses. The proposed time series model and its estimator make it possible to obtain consistent parameter estimates of a Markov chain and asymptotically normal esti- mates of transition rates, base intensities and parameter α, which controls “jumps” in intensity process.