Bayesian Analysis of Stochastic Process Models

1

Stochastic processes

1.1 Introduction

The theme of this book is Bayesian Analysis of Stochastic Process Models. In this first chapter, we shall provide the basic concepts needed in defining and analyzing stochastic processes. In particular, we shall review what stochastic processes are, their most important characteristics, the important classes of processes that shall be analyzed in later chapters, and the main inference and decision-making tasks that we shall be facing. We also set up the basic notation that will be followed in the rest of the book. This treatment is necessarily brief, as we cover material which is well known from, for example, the texts that we provide in our final discussion.

1.2 Key concepts in stochastic processes

Stochastic processes model systems that evolve randomly in time, space or space-time. This evolution will be described through an index . Consider a random experiment with sample space Ω, endowed with a σ-algebra and a base probability measure P. Associating numerical values with the elements of that space, we may define a family of random variables , which will be a stochastic process. This idea is formalized in our first definition that covers our ...