Caged Markov process – A continuous-time framework for modeling a constrained Markov process within a freely-evolving Markov process

We develop a continuous-time modeling framework for describing a bivariate Markov system in which the transition mechanism of one Markov process is constrained by another freely-evolving Markov process. The notion of constraint parameters is introduced to explicitly quantify the degree of constraints toward the constrained process within the coupling scheme. A complete construction of the resulting bivariate Markov model is provided with a closed-form formulation of its transition rate matrix. Several attributes of the model are examined, in particular, the behavior of the bivariate model when the two underlying Markov processes are independent from each other. This enables us to properly define a measurement of the overall strength of constraints based on the proposed constraint parameters. Model parameter estimation is discussed and demonstrated with two simulations, including a coupling of two independent Markov processes and a strongly coupled bivariate Markov process. The proposed model is further applied to real-world datasets in the context of stock index time series analysis.