A nonstandard 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)). AMS (MOS) subject classifications (1970). Primary 60H05, 60J65, 02H25.

Hyperfinite Dimensional Representations of Spaces and Algebras of Measures

“…However, it should be noted that not even all internal subsets of G ω are necessarily measurable with respect to λ f × λ g . Fortunately, the following fact is still true-see [4] or [1].…”

“…Besides the original Robinson's book [25], the standard general reference are the monographs [1] and [28]. For Loeb measure also the original papers by Loeb [18][19][20][21], as well as [4], [5], [13], [14] and the survey [6] by Cutland can be consulted. Some recent developments are reflected in [17], [3] and the survey [26] by Ross.…”

Hyperfinite Construction of G-Expectation

“…Anderson [2], Hoover and Perkins [9], Keisler [10], Lindstrøm [11], a few to mention, used Loeb's [12] approach to develop basic nonstandard stochastic analysis and in particular, the nonstandard Itô calculus. Loeb [12] also presents the construction of a Poisson processes using nonstandard analysis.…”

“…Loeb [12] also presents the construction of a Poisson processes using nonstandard analysis. Anderson [2] showed that Brownian motion can be constructed from a hyperfinite number of coin tosses, and provides a detailed proof using a special case of Donsker's theorem. Anderson [2] also gave a nonstandard construction of stochastic integration with respect to his construction of Brownian motion.…”

“…Anderson [2] showed that Brownian motion can be constructed from a hyperfinite number of coin tosses, and provides a detailed proof using a special case of Donsker's theorem. Anderson [2] also gave a nonstandard construction of stochastic integration with respect to his construction of Brownian motion. Keisler [10] uses Anderson's [2] result to obtain some results on stochastic differential equations.…”