Mathematical Preliminaries

Abstract

To prepare us for the subsequent study, this chapter summarizes certain background materials used in the rest of the book. Section 2.2 begins with the definitions of stochastic processes and filtrations, which lead to a very important concept in stochastic processes, namely, the notion of martingales. In Section 2.3, we recall the definition of Markov chains. Rather than working exclusively with their transition probabilities, this book concentrates on their generators. In view of various applications, it is practical and natural to characterize a Markov chain by using its generator. Given a generator, the construction of the associated Markov chain is described in Section 2.4 by means of the piecewise-deterministic process approach. Since one of the central themes of the book encompasses quasi-stationary distributions of singularly perturbed chains, we introduce this notion together with the weak and strong irreducibilities in Section 2.5, which are used extensively in the chapters to follow. Section 2.6 reviews Gaussian and diffusion processes. Finally, Section 2.7 closes the chapter with some postscript notes.