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This practical reference/text presents new computer methods of approximation, simulation, and visualization for a host of [alpha]-stable stochastic processes and shows how [alpha]-stable variates are useful in the modeling of various problems arising in economics, finances, chemistry, physics, and engineering - providing accurate descriptions of real phenomena.Offering detailed proofs for most of the results obtained, Simulation and Chaotic Behavior of [alpha]-Stable Stochastic Processes examines the properties of [alpha]-stable random variables and processes . . . supplies theoretical investigations and computer illustrations of the hierarchy of chaos for stochastic processes with applications to stochastic modeling . . . studies and characterizes the ergodic properties of different classes of stochastic processes . . . demonstrates how to apply the results obtained to a wide variety of disciplines . . . and more!With over 100 computer figures and carefully selected citations to the literature, Simulation and Chaotic Behavior of [alpha]-Stable Stochastic Processes is an invaluable reference for applied mathematicians and probabilists working with stochastic processes, chaos, stochastic modeling, discrete and approximate methods in stochastic analysis, and the applications of statistical methods; electrical and electronics engineers; financiers; physicists; biologists; and chemists; as well as an excellent text for upper-level undergraduate and graduate students in these disciplines taking courses in stochastic processes.

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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 L e' vy 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 L e' vy 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 It o ^ 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 L e' vy 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 L e' vy 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 Selt-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 L e' vy 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, L e' vy Flight, and L e' vy 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 It o ^ Integrals = 205
  8.3 The Rate of Convergence of LePage Type Series = 208
  8.4 Approximation of L e' vy α - Stable Diffusions = 217
  8.5 Applications to Statistical Tests of Hypotheses = 218
  8.6 L e' vy 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 Birkhoff 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