Conditional Markov Chains Revisited Part I: Construction and properties

In this paper we continue the study of conditional Markov chains (CMCs) with finite state spaces, that we initiated in Bielecki, Jakubowski and Niew\k{e}g{\l}owski (2014a) in an effort to enrich the theory of CMCs that was originated in Bielecki and Rutkowski (2004). We provide an alternative definition of a CMC and an alternative construction of a CMC via a change of probability measure. It turns out that our construction produces CMCs that are also doubly stochastic Markov chains (DSMCs), which allows for study of several properties of CMCs using tools available for DSMCs.