The Mathematics of Hypermodels

Siu-Ah Ng
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Siu-Ah Ng: School of Mathematical, Stats & IT, University Natal-Pietermaritzburg, South Africa

Chapter 7 in Hypermodels in Mathematical Finance:Modelling via Infinitesimal Analysis, 2003, pp 121-255 from World Scientific Publishing Co. Pte. Ltd.

Abstract: AbstractThe following sections are included:Mathematical logic and Classical HyperanalysisLogic and hypermodels *ℝConstruction of *ℝ using the compactness theoremConstruction *ℝ using ultrapowersSome basic properties of hypermodel *ℝHypermodels of ℝ in richer languagesSome examplesReferencesThe hyperuniverseDoing mathematics in the languages of set theoryHyperuniverse and the hyperextensionThe existence of the hyperextensionApplications and examplesReferencesHyperanalysis of probabilityThe Loeb measure constructionLoeb integration theorySome examplesHypermodels of Brownian MotionAnderson's discrete hypermodel of Brownian motionSome *continuous hypermodels of Brownian motionSome examplesItô Integral and Stochastic Differential EquationsSome general remarksWiener integral and Itô integralItô's lemma and the Stratonovitch integralSolving Stochastic differential equationsMalliavin calculusWhite noise analysisUniversality and homogeneity properties of hyperfinite models

Date: 2003
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