Simulation and Chaotic Behavior of Alpha-Stable Stochastic Processes Book

	Preface
1	Preliminary Remarks	1
1.1	Historical Overview	1
1.2	Stochastic [alpha]-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, [alpha]-Stable Levy Motion	9
2.2	Brownian Motion	9
2.3	The Poisson Process	20
2.4	[alpha]-Stable Random Variables	23
2.5	[alpha]-Stable Levy Motion	30
3	Computer Simulation of [alpha]-Stable Random Variables	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 [alpha]-Stable Distributions	47
3.6	Exact Formula for Tail Probabilities	51
3.7	Density Estimators	55
4	Stochastic Integration	67
4.2	Ito Stochastic Integral	69
4.3	[alpha]-Stable Stochastic Integrals of Deterministic Functions	73
4.4	Infinitely Divisible Processes	75
4.5	Stochastic Integrals with ID Integrators	79
4.6	Levy 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 [alpha]-Stable Levy Motion	107
5	Spectral Representations of Stationary Processes	111
5.2	Gaussian Stationary Processes	112
5.3	Representation of [alpha]-Stable Stochastic Processes	116
5.4	Structure of Stationary Stable Processes	127
5.5	Self-similar [alpha]-Stable Processes	134
6	Computer Approximations of Continuous Time Processes	141
6.2	Approximation of Diffusions Driven by Brownian Motion	142
6.3	Approximation of Diffusions Driven by [alpha]-Stable Levy Measure	156
6.4	Examples of Application in Mathematics	158
7	Examples of [alpha]-Stable Stochastic Modeling	171
7.1	Survey of [alpha]-Stable Modeling	171
7.2	Chaos, Levy Flight, and Levy 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
8.2	Error of Approximation of Ito Integrals	205
8.3	The Rate of Convergence of LePage Type Series	208
8.4	Approximation of Levy [alpha]-Stable Diffusions	217
8.5	Applications to Statistical Tests of Hypotheses	218
8.6	Levy 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
10.2	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