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 It integral or a proof of It's Lemma.