A Generalization of Separable Stochastic Processes

Abstract

Doob introduced the standard modifications or extensions of a stochastic process and proved that every stochastic process has a separable standard modification. In 1964 Elliott and Morse developed a general theory of product measures with implications in the theory of continuous parameter processes with mutually independent random variables. In particular, they gave a new method for obtaining extensions which considerably generalizes the notion of separability. For a separable process only certain events specified by restrictions of the random variables at a nondenumerable collection of time points are measurable. Under their generalization, the restriction to only certain events is virtually removed. The key to the new method for obtaining extensions is a modification by means of nilsets. The definition of nilsets has recently been adjusted to enable the application of this method to general stochastic processes.