Loeb Measures in Practice: Recent Advances: EMS Lectures 1997: Lecture Notes in Mathematics, cartea 1751
Autor Nigel J. Cutlanden Limba Engleză Paperback – 12 dec 2000
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Specificații
ISBN-13: 9783540413844
ISBN-10: 3540413847
Pagini: 132
Ilustrații: CXXXII, 118 p.
Dimensiuni: 155 x 235 x 7 mm
Greutate: 0.45 kg
Ediția:2000
Editura: Springer Berlin, Heidelberg
Colecția Springer
Seria Lecture Notes in Mathematics
Locul publicării:Berlin, Heidelberg, Germany
ISBN-10: 3540413847
Pagini: 132
Ilustrații: CXXXII, 118 p.
Dimensiuni: 155 x 235 x 7 mm
Greutate: 0.45 kg
Ediția:2000
Editura: Springer Berlin, Heidelberg
Colecția Springer
Seria Lecture Notes in Mathematics
Locul publicării:Berlin, Heidelberg, Germany
Public țintă
ResearchCuprins
1. Loeb Measures
1.1 Introduction
1.2 Nonstandard Analysis
1.2.1 The hyperreals
1.2.2 The nonstandard universe
1.2.3 N1-saturation
1.2.4 Nonstandard topology
1.3 Construction of Loeb Measures
1.3.1 Example: Lebesgue measure
1.3.2 Example: Haar measure
1.3.3 Example: Wiener measure
1.3.4 Loeb measurable functions 1.4 Loeb Integration Theory
1.5 Elementary Applications
1.5.1 Lebesgue integration
1.5.2 Peano's Existence Theorem
1.5.3 Ito integration and stochastic differential equations 2 Stochastic Fluid Mechanics
2.1 Introduction
2.1.1 Function spaces
2.1.2 Functional formulation of the Navier-Stokes equations
2.1.3 Definition of solutions to the stochastic Navier-Stokes equations
2.1.4 Nonstandard topology in Hilbert spaces
2.2 Solution of the Deterministic Navier-Stokes Equations
2.2.1 Uniqueness
2.3 Solution of the Stochastic Navier-Stokes Equations
2.3.1 Stochastic Flow
2.3.2 Nonhomogeneous stochastic Navier-Stokes equations
2.4 Stochastic Euler Equations
2.5 Statistical Solutions
2.5.1 The Foias equation
2.5.2 Construction of statistical solutions using Loeb measures
2.5.3 Measures by nonstandard densities
2.5.4 Construction of statistical solutions using nonstandard densities
2.5.5 Statistical solutions for stochastic Navier-Stokes equations
2.6 Attractors for the Navier-Stokes Equations
2.6.1 Introduction
2.6.2 Nonstandard attractors and standard attractors
2.6.3 Attractors for 3-dimensional Navier-Stokes equations
2.7 Measure Attractors for Stochastic Navier-Stokes Equations
2.8 Stochastic Attractors for Navier-Stokes Equations
2.8.1 Stochastic attractors
2.8.2 Existence of a stochastic attractor for the Navier-Stokes equations
2.9 Attractors for the 3-dimensional Stochastic Navier-Stokes Equations
3. Stochastic Calculus of Variations
3.1 Introduction
3.1.1 Notation
3.2 Flat Integral Representation of Wiener Measure
3.3 The Wiener Sphere
3.4 Brownian Motion on the Wiener Sphere and the Infinite Dimensional Ornstein-Uhlenbeck Process
3.5 Malliavin Calculus
3.5.1 Notation and preliminaries
3.5.2 The Wiener-Ito chaos decomposition
3.5.3 The derivation operator
3.5.4 The Skorohod integral
3.5.5 The Malliavin operator
4. Mathematical Finance Theory
4.1 Introduction
4.2 The Cox-Ross-Rubinstein Models
4.3 Options and Contingent Claims
4.3.1 Pricing a claim
4.4 The Black-Scholes Model
4.5 The Black-Scholes Model and Hyperfinite CRR Models
4.5.1 The Black-Scholes formula
4.5.2 General claims
4.6 Convergence of Market Models
4.7 Discretisation Schemes
4.8 Further Developments
4.8.1 Poisson pricing models
4.8.2 American options 4.8.3 Incomplete markets
4.8.4 Fractional Brownian motion
4.8.5 Interest rates
Index
1.1 Introduction
1.2 Nonstandard Analysis
1.2.1 The hyperreals
1.2.2 The nonstandard universe
1.2.3 N1-saturation
1.2.4 Nonstandard topology
1.3 Construction of Loeb Measures
1.3.1 Example: Lebesgue measure
1.3.2 Example: Haar measure
1.3.3 Example: Wiener measure
1.3.4 Loeb measurable functions 1.4 Loeb Integration Theory
1.5 Elementary Applications
1.5.1 Lebesgue integration
1.5.2 Peano's Existence Theorem
1.5.3 Ito integration and stochastic differential equations 2 Stochastic Fluid Mechanics
2.1 Introduction
2.1.1 Function spaces
2.1.2 Functional formulation of the Navier-Stokes equations
2.1.3 Definition of solutions to the stochastic Navier-Stokes equations
2.1.4 Nonstandard topology in Hilbert spaces
2.2 Solution of the Deterministic Navier-Stokes Equations
2.2.1 Uniqueness
2.3 Solution of the Stochastic Navier-Stokes Equations
2.3.1 Stochastic Flow
2.3.2 Nonhomogeneous stochastic Navier-Stokes equations
2.4 Stochastic Euler Equations
2.5 Statistical Solutions
2.5.1 The Foias equation
2.5.2 Construction of statistical solutions using Loeb measures
2.5.3 Measures by nonstandard densities
2.5.4 Construction of statistical solutions using nonstandard densities
2.5.5 Statistical solutions for stochastic Navier-Stokes equations
2.6 Attractors for the Navier-Stokes Equations
2.6.1 Introduction
2.6.2 Nonstandard attractors and standard attractors
2.6.3 Attractors for 3-dimensional Navier-Stokes equations
2.7 Measure Attractors for Stochastic Navier-Stokes Equations
2.8 Stochastic Attractors for Navier-Stokes Equations
2.8.1 Stochastic attractors
2.8.2 Existence of a stochastic attractor for the Navier-Stokes equations
2.9 Attractors for the 3-dimensional Stochastic Navier-Stokes Equations
3. Stochastic Calculus of Variations
3.1 Introduction
3.1.1 Notation
3.2 Flat Integral Representation of Wiener Measure
3.3 The Wiener Sphere
3.4 Brownian Motion on the Wiener Sphere and the Infinite Dimensional Ornstein-Uhlenbeck Process
3.5 Malliavin Calculus
3.5.1 Notation and preliminaries
3.5.2 The Wiener-Ito chaos decomposition
3.5.3 The derivation operator
3.5.4 The Skorohod integral
3.5.5 The Malliavin operator
4. Mathematical Finance Theory
4.1 Introduction
4.2 The Cox-Ross-Rubinstein Models
4.3 Options and Contingent Claims
4.3.1 Pricing a claim
4.4 The Black-Scholes Model
4.5 The Black-Scholes Model and Hyperfinite CRR Models
4.5.1 The Black-Scholes formula
4.5.2 General claims
4.6 Convergence of Market Models
4.7 Discretisation Schemes
4.8 Further Developments
4.8.1 Poisson pricing models
4.8.2 American options 4.8.3 Incomplete markets
4.8.4 Fractional Brownian motion
4.8.5 Interest rates
Index
Caracteristici
Includes supplementary material: sn.pub/extras