Conditional Markov chains: Properties, construction and structured dependence

In this paper we contribute to the theory of conditional Markov chains (CMCs) that take finitely many values and that admit intensity. We provide a method for constructing a CMC with given intensity and with given conditional initial law, and which is also a doubly stochastic Markov chain. We provide a martingale characterization for such process, and we discuss other useful properties. We define and give sufficient and necessary conditions for strong Markovian consistency and weak Markovian consistency of a multivariate CMC. We use these results to model structured dependence between univariate CMCs, that is, to model a multivariate CMC whose components are univariate CMCs with given laws. An example of potential application of our theory is presented.