Conversion from nonstandard to standard measure spaces and applications in probability theory

Abstract

ABSTRACT. Let (X, 3, v) be an internal measure space in a denumerably comprehensive enlargement.The set X is a standard measure space when equipped with the smallest standard o-algebra % containing the algebra a, where the extended real-valued measure p. on % is generated by the standard part of v. If / is fl-measurable, then its standard part / is jR-measurable on X. If / and p. are finite, then the vintegral of / is infinitely close to the /¿-integral of /. Applications include coin tossing and Poisson processes.In particular, there is an elementary proof of the strong Markov property for the stopping time of the ;'th event and a construction of standardsample functions for Poisson processes.

Hyperfinite Construction of G-Expectation

“…This approach is very promising because it also allows, for instance, to study 1 continuous-time stochastic processes as formally finite objects. Many authors have applied nonstandard analysis to problems in measure theory, probability theory and mathematical economics (see for example, Anderson and Raimondo [3] and the references therein or the contribution in Berg and Neves [4]), especially after Loeb [12] converted nonstandard measures (i.e. the images of standard measures under the nonstandard embedding * ) into real-valued, countably additive measures, by means of the standard part operator and Caratheodory's extension theorem.…”

“…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.…”

Near equivalence on metric spaces and a nonstandard central limit theorem

“…We had to develop a radically elementary model for studying weak convergence in metric spaces based on the metrics of Prohorov and Lévy (Section 2). A different notion of weak convergence (in topological, not simply metric, spaces) based on Robinson's nonstandard analysis [13] and Loeb measure spaces [9] is given by Anderson and Rachid [1]. Appendices A and B contain auxiliary material used in the paper.…”

Book Review: Radically elementary probability theory

“…Anderson [2] on the construction of Brownian motion. Keisler [5] on stochastic analysis, Loeb [6] on the construction of an important class of probability spaces (now called Loeb spaces), and Perkins' work [8] on Brownian local times. There are good expositions of ONSA, see especially [1 and 4] that provide both a manageable entry into the subject and a wealth of applications.…”