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.

In this chapter the concept of strong Markov consistency for multivariate Markov families and for multivariate Markov processes is introduced and studied. Strong Markov consistency of a multivariate Markov family/process, if satisfied, provides for invariance of the Markov property under coordinate projections, a property that is important in various practical applications. We only consider conservative Markov processes and Markov families.In Section 2.1, we study the so-called strong Markov consistency for multivariate Markov families and multivariate Markov processes taking values in an arbitrary metric space. This study is geared towards formulating a general framework within which the strong Markov consistency can be conveniently analyzed. In Section 2.2, we specify our study of the strong Markov consistency to the case of multivariate Feller-Markov families taking values in Rn. The analysis is first carried in the time-inhomogeneous case, and then in the time homogeneous case where a more comprehensive study can be done.

In this chapter we introduce and discuss various concepts of consistency for multivariate special semimartingales. The results here are mainly based on Theorem 5.1, which generalizes to the case of semimartingales that are not special. Thus, these results themselves generalize in a straightforward manner to the case of semimartingales that are not special. We chose to work with special semimartingales in order to ease somewhat the presentation. Throughout this chapter the semimartingale truncation functions will be considered to be standard truncation functions of appropriate dimensions. In what follows, the semimartingale characteristics will be always computed with respect to the relevant standard truncation functions. Thus, the semimartingale characteristics for all the semimartingales showing in the rest of this chapter are considered to be unique (as functions of the trajectories on the canonical space) once the filtration is chosen with respect to which the characteristics are computed. The theory is illustrated by various examples.

In this brief chapter we discuss the concept of semimartingale structure for a collection of special semimartingales. As in Chapter 5, we confine ourselves to the bivariate case only, and we consider semimartingale characteristics with respect to the standard truncation function. We start with definition of the semimartingale structure, and then we follow with examples.