Chapter 4 - Sample Function Analysis of Continuous Parameter Stochastic Processes

Abstract

This chapter discusses the sample function analysis of continuous parameter stochastic processes; and the concepts of continuity, differentiation, and integration in the L2 sense, have been introduced. The basic difficulty is that the finite-dimensional distributions do not yield enough information to determine the probability of such events. The basic approach would be to show that sample function analysis is possible for processes with special properties, namely, separability and measurability. A basic property of separable processes is that many sets whose definitions involve uncountably many values of the parameter t become measurable. Separability allows handling events and random variables that involve uncountably many values of the parameter t of a stochastic process. By the separability theorem, there is always a separable process having a specified consistent set of finite-dimensional distributions; in particular, there is a separable version of Brownian motion.