Conditional Markov Chains Revisited Part I: Construction and properties

TL;DR

This paper revisits conditional Markov chains with finite states, offering new definitions and constructions that connect them to doubly stochastic Markov chains, enhancing the theoretical framework and analytical tools available.

Contribution

It introduces an alternative definition and construction of CMCs via change of measure, linking them to DSMCs for deeper analysis.

Findings

Construction via change of measure produces CMCs that are also DSMCs

Enables study of CMC properties using DSMC tools

Extends the theoretical understanding of CMCs with finite states

Abstract

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.