A non-standard representation for Brownian Motion and Itô integration

Abstract

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} 7 *, 31 = {internal subsets of 12}, v(A) = \A \l2 n 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 Xfcco)-Z ^ + (i,r-[ l ,f])îi2ttl±l k<[nt] \nVf?and define 0: [0, 1] x 12 -• R by 0(f, co) = °x(t, co).

Star-finite representations of measure spaces

“…For example, Ward Henson [9] extended their procedure to represent essentially every finitely additive probability measure. Rohit Parikh and Milton Parnés [19,20] showed that the technique could be used to define the conditional probability P(A\B) for any pair of subsets of [0,1], retaining translation invariance.…”

“…Loeb [16] extracted standard harmonic measure on an ideal boundary and maximal representing measures for positive harmonic functions as distributions of internal measures. The author [1] constructed Wiener measure on C([0,1]) as the distribution of an internal measure v on *C([0,1]); this was possible even though there is a countable B G C([0,1]) such that v(*B) = 1. It is highly doubtful that either of these constructions could be carried out usefully without the use of measure-preserving maps.…”

“…In [1], the author constructed a Brownian motion as the standard part of a *-finite random walk x defined on a *-finite space fi. He then considered the problem of defining an Itô integral with respect to this Brownian motion.…”

“…This involves considering functions on Ü X [0,1]. Using the measure-preserving map st: *[0,1] -» [0, 1], any such function can be lifted to a function on Í2 X* [0,1]. In this setting, the special *-finite properties of x can be used to full advantage.…”

A Loeb-measure approach to theorems by Prohorov, Sazonov and Gross

“…It has been clear for some time now that the Loeb-measure of nonstandard analysis [11] may be useful in the construction of different kinds of limit measures. Indeed, Anderson's nonstandard construction of a Brownian motion [1] may be regarded as a direct construction of a weak limit measure (compare Billingsley [4]). Work on weak convergence from a nonstandard point of view has been carried on by Anderson and Rashid [3], and Loeb [12].…”