Abstract
In a recent paper [10], Peter A. Loeb showed how to convert non-standard measure spaces into standard ones and gave applications to probability theory. We apply these results to Brownian Motion and Itô integration. We first develop a number of new tools about Loeb spaces. We then show that Brownian Motion can be obtained as the Loeb process corresponding to a non-standard random walk obtained from a*-finite number of coin tosses. This permits a very constructive proof of a special case of Donsker's Theorem. The Itô integral with respect to this Brownian Motion is a non-standard Stieltjes integral with respect to the random walk. As a consequence, an easy proof of Itô’s Lemma is possible. The results in this paper were announced in [1].
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This work was carried out while the author was supported by a Canada Council Doctoral Fellowship. The author is grateful to Professors D. J. Brown, J. L. Doob, S. Kakutani, H. J. Keisler, P. E. Kopp, and P. A. Loeb for their helpful suggestions and criticisms.
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Anderson, R.M. A non-standard representation for Brownian Motion and Itô integration. Israel J. Math. 25, 15–46 (1976). https://doi.org/10.1007/BF02756559
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DOI: https://doi.org/10.1007/BF02756559