Statistical inference for mixed jump processes by Markov switching model with application to identify seismicity levels

Describing the non-stationarity of mixed stochastic jump processes presents a formidable challenge. This study employs a mixed compound Poisson process by Markov switching model (CPMSM) to address the issue of overdispersed data of random jumps. The Poisson rate is governed by a continuous-time Markov chain in this model. By assuming independence of the compound Poisson process and jump sizes, we present a generalized algorithm to estimate the parameters of CPMSM and try to predict the future behavior of this stochastic system. To investigate the potential application of this model and the accuracy of our algorithm, we provide a numerical simulation example in the context of the auto insurance payout scenario. The results suggest that CPMSM is an effective method of describing jump behavior controlled by random counts with mode switching via a Markov chain. Furthermore, we demonstrate a specific application of this model in identifying seismicity levels of Greek earthquake data and identify four hidden states. Finally, we compare its fitting effectiveness with the ETAS model using two public earthquake datasets in Japan and Iran.