Continuous-Parameter Martingales

Abstract

The second part of this book starts with a continuous-parameter extension of the discrete-parameter theory of Chapter 1. Our use of the term “extension” is quite misleading. Indeed, we will quickly find that in order to carry out these “extensions,” one needs a good understanding of the regularity of the sample functions of multiparameter stochastic processes; this will require a great effort. However, we will be rewarded for our hard work, since it will lead to a successful continuous-parameter theory that, in many ways, probes much more deeply than its discrete-parameter counterpart. Moreover, this theory lies at the foundations of nearly all of the random fields that arise throughout the rest of this book and a great deal more. Viewed as such, this chapter is simply indispensable for those who wish to read on.