Nonstandard Analysis

Theory and Applications

AvLeif O. Arkeryd,Nigel J. Cutland

Inbunden, Engelska, 1997

Del i serien Nato Science Series C:

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Beskrivning

Presenting a detailed introduction to the methodology of nonstandard analysis and the foundations of its use in analysis, topology, probability theory and stochastic analysis, this text includes articles concerning more advanced applications in functional analysis. These include stochastic differential equations, mathematical physics and mathematical finance theory.

Produktinformation

Utgivningsdatum:1997-04-30
Mått:170 x 244 x 27 mm
Vikt:842 g
Format:Inbunden
Språk:Engelska
Serie:Nato Science Series C:
Antal sidor:366
Upplaga:1997
Förlag:Kluwer Academic Publishers
ISBN:9780792345862

Utforska kategorier

Beräkning och matematisk analys inom Naturvetenskap och teknik

Innehållsförteckning

Foundations of Nonstandard Analysis A Gentle Introduction to Nonstandard Extensions.- 1 Introduction.- 2 Nonstandard Extensions.- 3 Logical Formulas.- 4 Nonstandard Extensions of Multisets.- 5 Nonstandard Extensions of the Multiset (X, P(X)).- 6 Superstructures.- 7 Saturation.- Nonstandard Real Analysis.- 1 Introduction.- 2 Basic Properties of *?.- 3 Sequences and Series.- 4 Continuity.- 5 Differentiation.- 6 Riemann Integration.- 7 Topology on ?.- 8 Using Internal Subsets of *?.- 9 An Application: Differential Equations.- Nonstandard Analysis and Topology.- 1 Metric and Topological Spaces.- 2 Continuous Mappings.- 3 Convergence.- 4 More on Topologies.- 5 Compact Spaces.- 6 Product Spaces.- 7 Restricted or Relative Topologies.- 8 Uniform Continuity on Metric Spaces.- 9 Nonstandard Hulls.- 10 Compactifications.- 11 More Exercises.- Loeb Measure and Probability.- 1 Introduction.- 2 Finite Loeb Measure.- 3 Constructing Standard Measures.- 4 Representing Standard Measures.- 5 Measurable Functions.- 6 Integration Theory.- 7 Probability Theory.- 8 Advertisement.- 9 Exercises.- An Introduction to Nonstandard Functional Analysis.- 1 Elementary Nonstandard Analysis of Normed Linear Spaces.- 2 Advanced Theory of Banach Spaces.- 3 Elementary Theory of Linear Operators.- 4 Spectral Theory of Bounded Operators.- 5 Applications of Nonstandard Spectral Theory.- 6 Closed Operators.- Applications of Nonstandard Analysis in Ordinary Differential Equations.- 1 Introduction.- 2 Tools in NSA.- 3 Differential Equations and Recursive Sequences.- 4 Regular Perturbations.- 5 Example.- 6 Dynamical Systems: Notions of Stability.- 7 Singular Perturbations.- Better Nonstandard Universes with Applications.- 1 Introduction.- 2 The Isomorphism Property.- 3 The Special Model Axiom and FullSaturation.- 4 The ?-Bolzano-Weierstrass Property.- Internal Martingales and Stochastic Integration.- 1 Hyperfinite Probability Spaces.- 2 Poisson Processes.- 3 Brownian Motion.- 4 Internal Martingales.- 5 Doob’s Inequality.- 6 Quadratic Variation.- 7 Standard Parts.- 8 S-continuity.- 9 Stochastic Integration.- 10 Itô’s Formula.- 11 Lévy’s Characterization of Brownian Motion.- 12 Connections to Standard Theory.- 13 Stochastic Integrals in Higher Dimensions.- 14 Stochastic Differential Equations.- 15 Brownian Local Time.- 16 The Infinite Dimensional Ornstein-Uhlenbeck Process.- Stochastic Differential Equations with Extra Properties.- 1 Introduction.- 2 Spaces of Stochastic Processes.- 3 Solutions of Stochastic Differential Equations.- 4 Solutions which are Markov Processes.- 5 A Fixed Point Theorem.- 6 Stochastic Differential Equations with Nondegenerate Coefficients.- Hyperfinite Mathematical Finance.- 1 Introduction.- 2 Finite Market Models.- 3 Pricing Options in a Hyperfinite CRR Model.- 4 Hyperfinite Trading Strategies.- 5 Convergence of Prices and Strategies.- 6 Further Developments.- Applications of Nsa to Mathematical Physics.- 1 A Kinetic Inequality.- 2 The Time Asymptotic Behaviour for Certain Rarefied Gases when the Incoming Fluxes at the Boundary are Given.- 3 On Semiclassical Limits for the Schrödinger Equation.- A Nonstandard Approach to Hydromechanics Navier-Stokes Equations.- 1 Introduction.- 2 Deterministic Navier-Stokes Equations.- 3 Statistical Solutions.- 4 Stochastic Equations.- 5 Some Open Problems.