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.

Modeling infinitely many agents

“…Below, we show that the existence of a pure-strategy Nash equilibrium can still be obtained when the nowhere equivalence condition is imposed on the agent space; such a result extends the theorem of Rauh (2007). 35 Such probability spaces were introduced by Loeb (1975). For the construction, see Loeb and Wolff (2015).…”

Equilibrium in Continuous-Time Financial Markets: Endogenously Dynamically Complete Markets

“…Let (Ω F L(μ)) be the (complete) Loeb measure generated by (Ω F μ) (Loeb (1975)). Although (Ω F L(μ)) is generated by a nonstandard construction, Loeb showed that it is a probability space in the usual standard sense.…”

“…We then use nonstandard analysis to produce a candidate equilibrium in the continuous-time model, show that the equilibrium in the hyperfinite model is infinitely close to the candidate equilibrium in the continuous-time model, verify that the candidate prices are dynamically complete, and are in fact equilibrium prices. Anderson (1976) provided a construction for Brownian motion and Brownian stochastic integration using Loeb (1975) measure-a measure in the usual standard sense produced by a nonstandard construction. Anderson's Brownian motion is a hyperfinite random walk which can simultaneously be viewed as being a standard Brownian motion in the usual sense of probability theory.…”

Internal Controls and Relaxed Controls

“…" eN Fix H 6 * N \ N ; it will be convenient to assume that H = 4 L for some L. Let At = H~l and let T = {0, At, 2At,..., HAt = 1}. We assume familiarity with the uniform Loeb measure P on T (see [11] or [9]). Now P is a standard probability measure on the <r-algebra <y({B : B £ T, B internal}), whose P-completion is denoted by L(T).…”

“…In this paper we show that under fairly general conditions on / , internal controls U correspond exactly to relaxed (or generalised) controls (see below for definition) in the following sense: given an internal control U there is a (unique) relaxed control v producing the same trajectory as U, and vice versa. The proof uses hyperfinite Loeb measure [11]. Relaxed controls were introduced by J. Warga [14] and a comprehensive account of them is provided in [15].…”