Nonstandard Analysis

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.