A new class of conditional Markov jump processes with regime switching and path dependence: properties and maximum likelihood estimation

This paper develops a new class of conditional Markov jump processes with regime switching and paths dependence. The key novel feature of the developed process lies on its ability to switch the transition rate as it moves from one state to another with switching probability depending on the current state and time of the process as well as its past trajectories. As such, the transition from current state to another depends on the holding time of the process in the state. Distributional properties of the process are given explicitly in terms of the speed regimes represented by a finite number of different transition matrices, the probabilities of selecting regime membership within each state, and past realization of the process. In particular, it has distributional equivalent stochastic representation with a general mixture of Markov jump processes introduced in Frydman and Surya (2020). Maximum likelihood estimates (MLE) of the distribution parameters of the process are derived in closed form. The estimation is done iteratively using the EM algorithm. Akaike information criterion is used to assess the goodness-of-fit of the selected model. An explicit observed Fisher information matrix of the MLE is derived for the calculation of standard errors of the MLE. The information matrix takes on a simplified form of the general matrix formula of Louis (1982). Large sample properties of the MLE are presented. In particular, the covariance matrix for the MLE of transition rates is equal to the Cram\'er-Rao lower bound, and is less for the MLE of regime membership. The simulation study confirms these findings and shows that the parameter estimates are accurate, consistent, and have asymptotic normality as the sample size increases.