In this chapter the concept of strong Markov consistency and the concept of weak Markov consistency for finite time-inhomogeneous multivariate Markov chains is 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.

Here we study the problem of constructing multivariate finite Markov chains whose coordinates are finite univariate Markov chains with given generator matrices. Specifically, we will be concerned here with construction of strong and weak Markov chain structures for a collection of finite Markov chains. We will use methods that are specific for Markov chains, and that are based on the results derived in Chapter 3. In this chapter we shall additionally be concerned with constructing weak Markov chain structures, which are related to the concept of weak Markov. Markov chain structures are key objects of interest in modeling structured dependence of Markovian type between stochastic dynamical given in terms of Markov chains. Accordingly, much of the discussion presented in this chapter is devoted to construction of Markov chain structures. Our construction allows for accommodating in a Markov structure model various dependence structures exhibited by phenomena one wants to model.

The Archimedean Survival Process (ASP), which is quite interesting from a theoretical point of view, originates in some financial applications. It turns out that applications of ASP and ASP structures go beyond finance. ASPs are very interesting objects to study in the context of stochastic structures, both from the theoretical and applied perspective.

A very interesting class of stochastic processes was introduced by Alan Hawkes (1971). These processes, now called Hawkes processes, are meant to model self-exciting and mutually-exciting random phenomena that evolve in time. The self-exciting phenomena are modeled as univariate Hawkes processes, and the mutually-exciting phenomena are modeled as multivariate Hawkes processes. Hawkes processes belong to the family of marked point processes, and, of course, a univariate Hawkes process is just a special case of the multivariate one. In this chapter we define and study generalized multivariate Hawkes processes, as well as the related consistencies and structures. Generalized multivariate Hawkes processes are multivariate marked point processes that add an important feature to the family of (classical) multivariate Hawkes processes: they allow for explicit modeling of simultaneous occurrence of excitation events coming from different sources, i.e. caused by different coordinates of the multivariate process.

In this chapter we extend the theory of Markov structures from the universe of classical (finite) Markov chains to the universe of (finite) conditional Markov chains. As it turns out such extension is not a trivial one. But, it is quite important both from the mathematical point of view and from the practical point of view. We will first discuss the strong conditional Markov chain structures, and then we will study the concept of the weak conditional Markov chain structures.