Hyperfinite stochastic integration for Lévy processes with finite-variation jump part

This article links the hyperfinite theory of stochastic integration with respect to certain hyperfinite Lévy processes with the elementary theory of pathwise stochastic integration with respect to pure-jump Lévy processes with finite-variation jump part. Since the hyperfinite Itô integral is also defined pathwise, these results show that hyperfinite stochastic integration provides a pathwise definition of the stochastic integral with respect to Lévy jump-diffusions with finite-variation jump part. As an application, we provide a short and direct nonstandard proof of the generalized Itô formula for stochastic differentials of smooth functions of Lévy jump-diffusions whose jumps are bounded from below in norm.Lévy processes, stochastic integration, nonstandard analysis, Itô formula

Everettian mechanics with hyperfinitely many worlds

The present paper shows how one might model Everettian quantum mechanics using hyperfinitely many worlds. A hyperfinite model allows one to consider idealized measurements of observables with continuous-valued spectra where different outcomes are associated with possibly infinitesimal probabilities. One can also prove hyperfinite formulations of Everett's limiting relative-frequency and randomness properties, theorems he considered central to his formulation of quantum mechanics. This approach also provides a more general framework in which to consider no-collapse formulations of quantum mechanics more generally.Comment: 22 pages. First version; comments very welcome

Nonstandard analysis for G-Stochastic calculus

Fadina TR. Nonstandard analysis for G-Stochastic calculus. Bielefeld: Bielefeld University; 2015.This thesis consists of three self-contained essays and a concluding chapter. The summary of the main results of the essays is the following: First, we prove a Donsker result for the G-Brownian motion with finite state-space. In the second essay, we give an elementary and more intuitive introduction to nonstandard measure theory and we also provide an alternative construction of the renowned Loeb measure. Following, we develop the basic theory for the hyperfinite G-expectation

Model theory and Ultrapower Embedding Problems in Operator Algebras

We survey the model theoretic approach to a variety of ultrapower embedding problems in operator algebras.Comment: 30 pages; first draft; comments welcome! To appear in the upcoming volume "Model theory of operator algebras" and the article contains references to other articles in the volum

Fra\"iss\'e limits of C*-algebras

We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II$_{1}$ factor as Fra\"iss\'e limits of suitable classes of structures. Moreover by means of Fra\"iss\'e theory we provide new examples of AF algebras with strong homogeneity properties. As a consequence of our analysis we deduce Ramsey-theoretic results about the class of full-matrix algebras.Comment: 19 pages. Final submitted versio

Hyperfinite construction of G-expectation

The hyperfinite G-expectation is a nonstandard discrete analogue of G-expectation (in the sense of Robinsonian nonstandard analysis). A lifting of a continuous-time G-expectation operator is defined as a hyperfinite G-expectation which is infinitely close, in the sense of nonstandard topology, to the continuous-time G-expectation. We develop the basic theory for hyperfinite G-expectations and prove an existence theorem for liftings of (continuous-time) G-expectation. For the proof of the lifting theorem, we use a new discretization theorem for the G-expectation (also established in this paper, based on the work of Dolinsky et al. [Weak approximation of G-expectations, Stoch. Process. Appl. 122(2) (2012), pp. 664–675])

Borel reducibility and classification of von Neumann algebras

We announce some new results regarding the classification problem for separable von Neumann algebras. Our results are obtained by applying the notion of Borel reducibility and Hjorth's theory of turbulence to the isomorphism relation for separable von Neumann algebras