In this chapter we study a family of adapted spaces that has been widely and successfully used in the nonstandard approach to stochastic analysis, the hyperfinite adapted spaces. The results in the monograph “An Infinitesimal Approach to Stochastic Analysis”, Keisler [1984], prompted a natural question: Why are these spaces so “rich” or “well behaved”?

In order to answer this question, we built a probability logic adequate for the study stochastic processes: adapted probability logic (see Keisler [1979], Keisler [1985], Keisler [1986a], Keisler [1986b], Hoover and Keisler [1984], Fajardo [1985a]). This is the origin of the theory we are describing in this book. We chose a somewhat different approach in Chapter 1 in order to introduce the theory in a smooth way without any need for a background in logic.

Basic nonstandard probability theory is a necessary prerequisite for most of this chapter. This theory is readily available to the interested mathematician without going through the technical literature on nonstandard analysis (see, among others, Albeverio, Fenstad, Hoegh-Krohn, and Lindstrom [1986], Cutland [1983], Fajardo [1990b], Lindstrom [1988], Stroyan and Bayod [1986] andKeisler [1988]). Nonetheless, in the following section we collect the main definitions and results needed in this book.

This seems to be an appropriate place to add a remark about the use of nonstandard analysis. It has been very hard to convince the mathematical community of the attractive characteristics of nonstandard analysis and its possible uses as a mathematical tool. This chapter, among other things, continues the task of showing with direct evidence the enormous potential that we believe nonstandard analysis has to offer to mathematics. The paper Keisler [1994] examines some of the reasons why nonstandard analysis has developed in the way we know it today and discusses the perspectives and possibilities in the years to come.

In this chapter we will establish additional properties of rich adapted spaces which can be applied to stochastic analysis. In Sections 8.1 and 8.2 we give an overview of the theory of neometric spaces, culminating in the Approximation Theorem. Section 8.C gives some typical applications of this theorem.

We briefly survey the evolution of the ideas in this chapter. The paper “From discrete to continuous time”, Keisler [1991], introduced a forcing procedure, resembling model theoretic forcing (see Hodges [1985]), which reduced statements about continuous time processes to approximate statements about discrete time processes without going through the full lifting procedure. After some refinements, in the series of papers Fajardo and Keisler [1996a] – Fajardo and Keisler [1995] we worked out a new theory, called the theory of neometric spaces, which has the following objectives: “First, to make the use of nonstandard analysis more accessible to mathematicians, and second, to gain a deeper understanding of why nonstandard analysis leads to new existence theorems. The neometric method is intended to be more than a proof technique—it has the potential to suggest new conjectures and new proofs in a wide variety of settings.” (From the introduction in Keisler [1995]).

This theory developed an axiomatic framework built around the notion of a neometric space, which is a metric space with a family of subsets called neocompact sets that are, like the compact sets, closed under operations such as finite union, countable intersection, and projection. In particular, for each adapted probability space there is an associated neometric space of random variables, and the adapted space is rich if the family of neocompact sets is countably compact.

In this book we take advantage of the results in the more recent paper Keisler [1997a] to give a simpler approach to the subject. In the Chapter 7 we defined rich adapted spaces directly without introducing neocompact sets. In this chapter we will give a quick overviewof the theory of neometric spaces. The neocompact sets will be defined here as countable intersections of basic sections, and their closure properties will be proved as theorems which hold in any rich adapted space.

This book studies stochastic processes using ideas from model theory. Some key tools come from nonstandard analysis. It is written for readers from each of these three areas. We begin by intuitively describing this work from each of the three viewpoints.

From the viewpoint of probability theory, this is a general study of stochastic processes on adapted spaces based on the notion of adapted distribution. This notion is the analog for adapted spaces of the finite dimensional distribution, and was introduced by Hoover and Keisler [1984]. It gives us a way of comparing stochastic processes even if they are defined on different adapted spaces. Acentral theme will be the question

When are two stochastic processes alike?

There are several possible answers depending on the problem at hand, but our favorite answer is: Two stochastic processes are alike if they have the same adapted distribution. Early on in this book, we will consider questions of the following kind about an adapted space, with the above meaning of the word “alike”.

1. (1) Given a stochastic process x on some other adapted space, will there always be a process like x on?

2. (2) If a problem with processes on as parameters has a weak solution, will it have a solution on with respect to the original parameters?

3. (3) If two processes x, y on are alike, is there is an automorphism of which preserves measures and filtrations and sends x to y?

Questions (1) – (3) ask whether an adapted space is rich enough for some purpose. Adapted spaces with these properties are said to be universal, saturated, and homogeneous, respectively. Several arguments in probability theory can be simplified by working with a saturated adapted space, especially existence theorems which ordinarily require a change in the adapted space. In practice, probability theory allows great freedom in the choice of the adapted space. One does not care much which space is being used, as long as it is rich enough to contain the processes of interest.

It is fair to say that the acceptability of impredicative definitions and reasoning in mathematics is not now, and hasn't been in recent years, a matter of major controversy. Solomon Feferman may regret that state of affairs, even though his ownwork contributed a great deal to bringing it about. I see thework of Feferman andKurt Schütte on the analysis of predicative provability in the 1960's as bringing to closure one aspect of the discussion of predicativity that began with Poincaré ‘s protests against “non-predicative definitions” in the first decade of the twentieth century and with Russell's making a “vicious circle principle” a major principle by which constructions in logic should be assessed. Although in the 1950's Paul Lorenzen and Hao Wang had undertaken to reconstruct mathematics in such a way that impredicativitywould be avoided, insistence on this (to which evenWang did not subscribe) was very much a minority view, and Feferman in particular sought principally to analyze what predicativity is, with the understanding that some aspects of this enterprise would require impredicative methods. The picture has changed since then by work to which he has also contributed, which has brought to light how much of classical analysis in particular can be done by methods that are logically very weak, in particular predicative.

The pioneer of this latter effort, as Feferman has analyzed in detail, was Hermann Weyl. Weyl also brought about the most dramatic episode in the early history by claiming that there is a “vicious circle” of the kind pointed to by Poincaré and Russell in some basic reasonings in analysis. Curiously, his promising beginning for a predicative reconstruction of analysis was not pursued further at the time either by him or by others. It may seem that Weyl's charge of a vicious circle found few adherents, but this appearance may be misleading because intuitionist analysis, which was just then being developed seriously by Brouwer, was not thought to be subject to the same difficulty. For some time thereafter, however, workers in foundations who accepted classical mathematics thought it necessary to reply to Weyl.