Bayesian Analysis of Stochastic Process Models

6

Continuous time continuous space processes

6.1 Introduction

Many processes have been proposed to model events in continuous time. Interest in continuous time models has been motivated not only by mathematical and methodological reasons, but has also been strongly influenced by applications in areas such as finance, telecommunications, and environmental sciences, to name just a few fields. The value of a stock option or the capital of an insurance company subject to premium payments and insurers’ claims, the number of packets transferred in asynchronous transfer mode (ATM), or the ozone level in a region are examples of phenomena that can be modeled with continuous time processes. Two simple, widely used examples of continuous time processes have already been studied earlier in this book. These are continuous time Markov chains, which were examined in Chapter 4, and Poisson processes, which were illustrated in Chapter 5. In this chapter we shall present other continuous time processes over continuous-state spaces, providing the basic properties and some examples. In particular, we shall consider Gaussian processes in Section 6.2, illustrating their inference and use as emulators of computer code simulators. A special case of the Gaussian process, the Brownian motion (or Wiener process) are presented in Section 6.3, along with fractional Brownian motion (FBM). Then, diffusions are the subject of Section 6.4 and a predator–prey model is presented in Section 6.5, as an example of ...