TABLE OF CONTENTS Using Andersen s hypermodel of Brownian motion, we show that It integral is the standard part of a hyperfinite sum, hence It integral can be realized as a Riemann sum . It s lemma is proved. This gives an integral representation of a functional of Brownian motion. Stratonovitch integral form of this lemma is also proved. Keisler used the hyperfinite representation of It integral to convert stochastic differential equations to hyperfinite difference equations, and solutions are obtained by hyperfinite iteration. We prove a special case of his general results as a demonstration of these methods. The main part of these methods involve construction of the appropriate adapted liftings to functions on some adapted hyperfinite probability space. These spaces are very well-behaved and rich in the sense that they are homogeneous and universal These are not available to most classical spaces. A result of Keisler s shows that in a formal sense, whatever happens in some space happens on an adapted hyperfinite probability space. This explains the reasons why solutions obtained by hypermodel methods are strong solutions.
Most of these results generalize to martingales as well, but we prefer to deal only with the Brownian case to give a more elegant presentation.
We first fix some notation throughout this section. We continue to let 
denote the hyperfinite time line, and ? the normalized counting measure on \{1}, ?({1})=0. So for 
,
Unless specified, ( ?, A, ?)...
