Thesis (Ph. D.)--University of Wisconsin--Madison, 1980. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 409-443).

The investigation is initially a continuation of Neuberger's work on linear boundary value problems. A very general iterative procedure for solution of these problems is described. The alternating-projection theorem of von Neumann is the mathematical starting point for this study. Later theorems demonstrate the validity of numerical approximation for Neuberger's method under certain conditions. A sampling of differential equations within the scope of our iterative method is given. The numerical evidence is that the procedure works well on neutral-state equations, for which no software is written now.

This thesis provides a study on stochastic models of financial markets related to problems of asset pricing and hedging, optimal portfolio managing and statistical changepoint detection in trends of asset prices. Chapter 1 develops a general model of a system of interconnected stochastic markets associated with a directed acyclic graph. The main result of the chapter provides sufficient conditions of hedgeability of contracts in the model. These conditions are expressed in terms of consistent price systems, which generalise the notion of equivalent martingale measures. Using the general results obtained, a particular model of an asset market with transaction costs and portfolio constraints is studied. In the second chapter the problem of multi-period utility maximisation in the general market model is considered. The aim of the chapter is to establish the existence of systems of supporting prices, which play the role of Lagrange multipliers and allow to decompose a multi-period constrained utility maximisation problem into a family of single-period and unconstrained problems. Their existence is proved under conditions similar to those of Chapter 1.The last chapter is devoted to applications of statistical sequential methods for detecting trend changes in asset prices. A model where prices are driven by a geometric Gaussian random walk with changing mean and variance is proposed, and the problem of choosing the optimal moment of time to sell an asset is studied. The main theorem of the chapter describes the structure of the optimal selling moments in terms of the Shiryaev–Roberts statistic and the posterior probability process.

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In this thesis are shown some results of realization of refinement monoid by Grothendieck monoids of von Neumann regular ring. The most important result of this thesis, recently published by P. Ara, J. Bosa and E. Pardo, claims that every finitely generated conical refinement monoid is realizable. We construct for each such monoid a von Neumann regular algebra such that monoid is realizable by this algebra. For this construction we use adaptable separated graphs and their relation to refinement monoids. 1

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yes / The properties of quantum probabilities are linked to the geometry of quantum mechanics, described by the Birkhoff-von Neumann lattice. Quantum probabilities violate the additivity property of Kolmogorov probabilities, and they are interpreted as Dempster-Shafer probabilities. Deviations from the additivity property are quantified with the Möbius (or non-additivity) operators which are defined through Möbius transforms, and which are shown to be intimately related to commutators. The lack of distributivity in the Birkhoff-von Neumann lattice Λd, causes deviations from the law of the total probability (which is central in Kolmogorov’s probability theory). Projectors which quantify the lack of distributivity in Λd, and also deviations from the law of the total probability, are introduced. All these operators, are observables and they can be measured experimentally. Constraints for the Möbius operators, which are based on the properties of the Birkhoff-von Neumann lattice (which in the case of finite quantum systems is a modular lattice), are derived. Application of this formalism in the context of coherent states, generalizes coherence to multi-dimensional structures.

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