ISBN: 3540413847
TITEL: Lecture Notes in Mathematics, Vol. 1751
AUTHOR: 
TOC:

1. Loeb Measures 1
1.1 Introduction 1
1.2 Nonstandard Analysis 2
1.2.1 The hyperreals 2
1.2.2 The nonstandard universe 7
1.2.3 Nr-saturation 10
1.2.4 Nonstandard topology 11
1.3 Construction of Loeb Measures 13
1.3.1 Example: Lebesgue measure 16
1.3.2 Example: Haar measure 17
1.3.3 Example: Wiener measure 17
1.3.4 Loeb measurable functions 19
1.4 Loeb Integration Theory 20
1.5 Elementary Apphcations 23
1.5.1 Lebesgue integration 23
1.5.2 Peano's Existence Theorem 24
1.5.3 Itintegration and stochastic differential equations 27
2. Stochastic Fluid Mechanics 29
2.1 Introduction 29
2.1.1 Function spaces 30
2.1.2 Functional formulation of the Navier-Stokes equations 31
2.1.3 Definition of solutions to the stochastic Navier-Stokes quations 32
2.1.4 Nonstandard topology in Hilbert spaces 33
2.2 Solution of the Deterministic Navier-Stokes Equations 34
2.2.1 Uniqueness 36
2.3 Solution of the Stochastic Navier-Stokes Equations 37
2.3.1 Stochastic flow 39
2.3.2 Nonhomogeneous stochastic Navier-Stokes equations 40
2.4 Stochastic Euler Equations 40
2.5 Statistical Solutions 42
2.5.1 The Foias equation 42
2.5.2 Construction of statistical solutions using Loeb measures 43
2.5.3 Measures by nonstandard densities 44
2.5.4 Construction of statistical solutions using nonstandard densities 45
2.5.5 Statistical solutions for stochastic Navier-Stokes equations 46
2.6 Attractors for Navier-Stokes Equations 46
2.6.1 Introduction 46
2.6.2 Nonstandard attractors and standard attractors 48
2.6.3 Attractors for 3-dimensional Navier-Stokes equations 49
2.7 Measure Attractors for Stochastic Navier-Stokes Equations 49
2.8 Stochastic Attractors for Navier-Stokes Equations 50
2.8.1 Stochastic attractors 52
2.8.2 Existence of a stochastic attractor for the Navier-Stokes equations 53
2.9 Attractors for 3-dimensional Stochastic Navier-Stokes Equations 55
Stochastic Calculus of Variations 61
3.1 Introduction 61
3.1.1 Notation 63
3.2 Flat Integral Representation of Wiener Measure 64
3.3 The Wiener Sphere 66
3.4 Brownian Motion on the Wiener Sphere and the Infinite Dimensional Ornstein-Uhlenbeck Process 69
3.5 Malliavin Calculus 72
3.5.1 Notation and preliminaries 73
3.5.2 The Wiener-It6 chaos decomposition 75
3.5.3 The derivation operator 77
3.5.4 The Skorohod integral 79
3.5.5 The Malliavin operator 83
4. Mathematical Finance Theory 85
4.1 Introduction 85
4.2 The Cox-Ross-Rubinstein Models 86
4.3 Options and Contingent Claims 88
4.3.1 Pricing a claim 90
4.4 The Black-Scholes Model 92
4.5 The Black-Scholes Model and Hyperfinite CRR Models 94
4.5.1 The Black-Scholes formula 95
4.5.2 General claims 95
4.6 Convergence of Market Models 96
4.7 Discretisation Schemes 98
4.8 Further Developments 99
4.8.1 Poisson pricing models 99
4.8.2 American options 99
4.8.3 Incomplete markets 100
4.8.4 Fractional Brownian motion 100
4.8.5 Interest rates 101
Index 109
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