In this chapter the concept of strong Markov consistency and the concept of weak Markov consistency for finite time-inhomogeneous multivariate Markov chainsis introduced and studied. In particular, necessary and sufficient conditions for both types of Markov consistency are given. The main tool used here is the semimartingale characterization of finite Markov chains. In addition, operator interpretation of a sufficient condition for strong Markov consistency and a necessary condition for weak Markov consistency are provided.By definition, strong Markov consistency implies the weak Markov consistency. In this chapter we provide sufficient condition for the reverse implication to hold.

Conditional Markov Chains are an important class of stochastic processes, and thus, study of the related consistency problems is important. Finite conditional Markov chains generalize classical finite Markov chains. Thus, in many ways, the study of Markov consistency for finite multivariate conditional Markov chains done in this chapter is a generalization of the study done in Chapter 3. In particular, the results derived here are nicely illustrated by their counterparts given in the simpler set-up of Chapter 3.