Markov Models

Abstract

Mathematically, stochastic hybrid systems can be described by Markov processes with discrete and continuous behaviours. This chapter provides in an incremental way the necessary mathematical background for understanding such complex systems. The chapter starts with discrete and continuous-time Markov chains. In this book, a Markov chain is understood as a stochastic process with the Markov property (or memoryless property, i.e. its future evolution depends only on the current state) defined on a discrete (finite or countable) state space. The Markov property is illustrated in the Chapman-Kolmogorov equation satisfied by the transition probabilities. Usually, in the continuous-time Markov case, the evolution of these transition probabilities is described by forward/backward Kolmogorov equations. These equations are expressed in terms of the stochastic matrix (called also infinitesimal generator) associated to the Markov chain. This is the matrix of transition rates and the practical use of Markov chains resides in our ability of handling it.

The second part of this chapter is dedicated to Markov processes defined on continuous state spaces that can be equipped with additional structures as sigma-algebras, metrics, topologies, norms or others like specific algebraic structures (lattice, vector space). Characterisations of such processes are usually given using functional analysis operators like: the operator semigroup/resolvent, the infinitesimal generator or other operators that can be associated to a Markov process. At an abstract level, these operators describe the evolution the evolution of transition probabilities for a Markov process. Chapman-Kolmogorov equation, Kolmogorov equations and other properties of these operators (e.g. Dynkin formula, martingale problem) represent the natural tools for studying such Markov processes.

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-1-4471-2795-6_12