Rich and Saturated Adapted Spaces

Abstract

In the paper [HK] the notions of an adapted distribution and of a saturated adapted probability space were introduced. The adapted distribution of a random variable on an adapted space (with values in a complete separable metric space) is the natural analogue of the distribution of a random variable on a probability space. An adapted space Ω is saturated if for any random variable x on Ω and pair of random variables x and ȳ on another adapted space Γ such that x and x have the same adapted distribution, there is a random variable y on Ω such that (x, y) and (x, ȳ) have the same adapted distribution. For stochastic differential equations and a wide variety of other existence problems, every existence theorem which holds on some adapted space holds on a saturated adapted space. The paper [FK1] introduced a new method for proving existence theorems in probability theory, based on the notion of a neocompact set of random variables. A set of random variables on an adapted space is said to be basic if it is either compact or is the set of all random variables which are measurable at time t and whose law belongs to a compact set C of measures, for some t and C. The family of neocompact sets is the closure of the family of basic sets under finite unions and Cartesian products, countable intersections, existential projections, and “universal