First Exit Time For Two Classes Of Hybrid Systems And Their Statistical Inference

This paper focuses on the mean exit time,scale functions,escape probability and parameter estimations of two classes of hybrid systems that are driven by a continuoustime Markov chain.First of all,under certain assumptions,according to the generator for the Markov process,the explicit expressions of the mean exit time,scale functions and escape probability of the hybrid geometric Brownian motion are given by solving the Poisson equations with some conditions.Furthermore,some explicit expressions for the mean exit time and escape probability in any boundary area（a,b）of symmetricα-stable processes are given.Then,the expressions of the parameter estimators of the hybrid geometric Brownian motion and the hybrid Cauchy process are given by the composite likelihood method and the EM algorithm.In addition,given the sequence of observations,we implement the Viterbi algorithm to find the most likely sequence of states for the Markov chain of the hybrid geometric Brownian motion.Finally,the hybrid geometric Brownian motion,the hybrid Cauchy process and Gaussian-HMM are used for the empirical analysis based on Shanghai composite index respectively.