# The Mixture of Markov Jump Processes: Monte Carlo Method and the EM Estimation

## TL;DR

This paper introduces a Monte Carlo method and EM algorithm for statistical estimation of a mixture of Markov jump processes with unobservable regimes, enabling better modeling of complex stochastic systems.

## Contribution

It provides a novel Monte Carlo simulation approach and an EM-based estimation procedure for a generalized mixture of Markov jump processes, extending previous models.

## Key findings

- Monte Carlo method accurately simulates the process.
- EM algorithm effectively estimates model parameters.
- Numerical examples demonstrate method performance.

## Abstract

This paper discusses tractable development and statistical estimation of a continuous time stochastic process with a finite state space having non-Markov property. The process is formed by a finite mixture of right-continuous Markov jump processes moving at different speeds on the same finite state space, whereas the speed regimes are assumed to be unobservable. The mixture was first proposed by Frydman (J. Am. Stat. Assoc., 100, 1046-1053, 2005) in 2005 and recently generalized in Surya (Stoch. Syst. 8, 29-44, 2018), in which distributional properties and explicit identities of the process are given in its full generality. The contribution of this paper is two fold. First, we present Monte Carlo method for constructing the process and show distributional equivalence between the simulated process and the actual process. Secondly, we perform statistical inference on the distribution parameters of the process. Under complete observation of the sample paths, maximum likelihood estimates are given in explicit form in terms of sufficient statistics of the process. Estimation under incomplete observation is performed using the EM algorithm. The estimation results completely characterize the process in terms of the initial probability of starting the process in any phase of the state space, intensity matrices of the underlying Markov jump processes, and the switching probability matrix of the process. Some numerical examples are given to test the performance of the developed method. The proposed estimation generalizes the existing statistical inferences for the Markov model by Albert (Ann. Math. Statist., 38, p.727-753., 1961), the mover-stayer model by Frydman (J. Am. Stat. Assoc.79, 632-638., 1984) and the Markov mixture model by Frydman (J. Am. Stat. Assoc., 100, 1046-1053, 2005).