Table of Contents
Preface iii
1 Preliminary Remarks 1
1.1 Historical Overview 1
1.2 Stochastic α-Stable Modeling 3
1.3 Statistical versus Stochastic Modeling 4
1.4 Hierarchy of Chaos 6
1.5 Computer Simulations and Visualizations 6
1.6 Stochastic Processes 7
2 Brownian Motion, Poisson Process, α-Stable Lévy Motion 9
2.1 Introduction 9
2.2 Brownian Motion 9
2.3 The Poisson Process 20
2.4 α-Stable Random Variables 23
2.5 α-Stable Lévy Motion 30
3 Computer Simulation of α-Stable Random Variables 35
3.1 Introduction 35
3.2 Computer Methods of Generation of Random Variables 36
3.3 Series Representations of Stable Random Variables 40
3.4 Convergence of LePage Random Series 43
3.5 Computer Generation of α-Stable Distributions 47
3.6 Exact Formula for Tail Probabilities 51
3.7 Density Estimators 55
4 Stochastic Integration 67
4.1 Introduction 67
4.2 Ito Stochastic Integral 69
4.3 α-Stable Stochastic Integrals of Deterministic Functions 73
4.4 Infinitely Divisible Processes 75
4.5 Stochastic Integrals with ID Integrators 79
4.6 Lévy Characteristics 83
4.7 Stochastic Processes as Integrators 86
4.8 Integrals of Deterministic Functions with ID Integrators 90
4.9 Integrals with Stochastic Integrands and ID Integrators 96
4.10 Diffusions Driven by Brownian Motion 101
4.11 Diffusions Driven by α-Stable Lévy Motion 107
5 Spectral Representations of Stationary Processes 111
5.1 Introduction 111
5.2 Gaussian Stationary Processes 112
5.3 Representation of α-Stable Stochastic Processes 116
5.4 Structure of Stationary Stable Processes 127
5.5 Self-similar α-Stable Processes 134
6 Computer Approximations of Continuous Time Processes 141
6.1 Introduction 141
6.2 Approximation of Diffusions Driven by Brownian Motion 142
6.3 Approximation of Diffusions Driven by α-Stable Lévy Measure 156
6.4 Examples of Application in Mathematics 158
7 Examples of α-Stable Stochastic Modeling 171
7.1 Survey of α-Stable Modeling 171
7.2 Chaos, Lévy Flight, and Lévy Walk 173
7.3 Examples of Diffusions in Physics 179
7.4 Logistic Model of Population Growth 192
7.5 Option Pricing Model in Financial Economics 196
8 Convergence of Approximate Methods 203
8.1 Introduction 203
8.2 Error of Approximation of Itô Integrals 205
8.3 The Rate of Convergence of LePage Type Series 208
8.4 Approximation of Lévy α-Stable Diffusions 217
8.5 Applications to Statistical Tests of Hypotheses 218
8.6 Lévy Processes and Poisson Random Measures 224
8.7 Limit Theorems for Sums of i.i.d. Random Variables 226
9 Chaotic Behavior of Stationary Processes 231
9.1 Examples of Chaotic Behavior 231
9.2 Ergodic Property of Stationary Gaussian Processes 239
9.3 Basic Facts of General Ergodic Theory 242
9.4 Birkboff Theorem for Stationary Processes 246
9.5 Hierarchy of Chaotic Properties 251
9.6 Dynamical Functional 255
10 Hierarchy of Chaos for Stable and ID Stationary Processes 263
10.1 Introduction 263
10.2 Ergodicity of Stable Processes 265
10.3 Mixing and Other Chaotic Properties of Stable Processes 279
10.4 Introduction to Stationary ID Processes 287
10.5 Ergodic Properties of ID Processes 295
10.6 Mixing Properties of ID Processes 297
10.7 Examples of Chaotic Behavior of ID Processes 302
10.8 Random Measures on Sequences of Sets 307
Appendix: A Guide to Simulation 315
Bibliography 339
Index 353
