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.

It is argued in this chapter that stochastic structures constitute a versatile tool that has many practical applications. Some of these applications have already been worked out, and some others are still to be worked out. In this chapter we provide a survey of existing applications of stochastic structures, and we also suggest some new potential applications.

In this chapter we define and study strong Markov family structures for a collection of time-homogeneous nice R-Feller–Markov families. Markov structures are key objects of interest in modeling structured dependence of Markovian type between stochastic dynamical systems of Markovian type, such as Markov families or Markov processes. Much of the discussion presented in this chapter is devoted to construction of Markov structures. Part of the input to any respective construction procedure is provided by marginal data, which we refer to as marginal inputs. Another part of the input is provided by data and/or postulates regarding stochastic dependence between the coordinates of the resulting Markov structure, which we refer to as dependence structure input. These inputs have to be appropriately accounted for in constructions of Markov structures. This, in principle, can be done, since, as discussed in this chapter, one has quite substantial flexibility in constructing Markov structures, which allows for accommodating in a Markov structure model various dependence structures exhibited by phenomena one wants to model.

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.