A Nonstandard Representation for Brownian Motion and Itô Integration

A number of authors have attempted to apply Nonstandard Analysis to Probability Theory. Unfortunately, the nonstandard reformulations heretofore proposed have retained most of the essential difficulties inherent in the standard formulations. As a result, the application of nonstandard techniques has met with limited success. Hersh [4] produced a nonstandard analogue of Wiener measure. His "measure", however, is not countably additive; moreover, it is supported on a countable subset of C([0, 1]). Using a different approach, Hersh and Greenwood [5] established some interesting results about nonstandard increments in Brownian motion and other stochastic processes, but failed to produce a successful formulation of the Itô integral or a proof of Itô's Lemma. In a recent paper [9], Peter A. Loeb introduced a new technique for formulating probabilistic problems in nonstandard terms. He showed that any nonstandard measure space within a denumerably comprehensive enlargement could be converted into a standard measure space which inherited important structural properties from the nonstandard space. Loeb gave applications to coin tossing and the Poisson process. The present paper outlines how Brownian motion and Itô integration can be successfully treated using Loeb's technique; the details will be presented in a subsequent article. Let 7? be an infinite natural number, in the sense of nonstandard analysis. Define an internal measure space (£2, 21, v) by £2 = {-1 , l}*, 31 = {internal subsets of 12}, v(A) = \A \l2 for A G 21. Thus, v is counting measure. Loeb's results show that the standard part of v is necessarily countably additive; hence, by the Carathéodory Extension Theorem, it has a unique extension to the a-field generated by 21. We shall call the completion of this extension the Loeb space corresponding to (12, 21, v), and denote it by (12,1(21), L(y)). Define a random walk x*[0, 1] x 12 —> *R by