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The modeling of stochastic dependence is fundamental for understanding random systems evolving in time. When measured through linear correlation, many of these systems exhibit a slow correlation decay--a phenomenon often referred to as long-memory or long-range dependence. An example of this is the absolute returns of equity data in finance. Selfsimilar stochastic processes (particularly fractional Brownian motion) have long been postulated as a means to model this behavior, and the concept of selfsimilarity for a stochastic process is now proving to be extraordinarily useful. Selfsimilarity translates into the equality in distribution between the process under a linear time change and the same process properly scaled in space, a simple scaling property that yields a remarkably rich theory with far-flung applications.
After a short historical overview, this book describes the current state of knowledge about selfsimilar processes and their applications. Concepts, definitions and basic properties are emphasized, giving the reader a road map of the realm of selfsimilarity that allows for further exploration. Such topics as noncentral limit theory, long-range dependence, and operator selfsimilarity are covered alongside statistical estimation, simulation, sample path properties, and stochastic differential equations driven by selfsimilar processes. Numerous references point the reader to current applications.
Though the text uses the mathematical language of the theory of stochastic processes, researchers and end-users from such diverse fields as mathematics, physics, biology, telecommunications, finance, econometrics, and environmental science will find it an ideal entry point for studying the already extensive theory and applications of selfsimilarity.
| Preface | p. ix |
| Introduction | p. 1 |
| Definition of Selfsimilarity | p. 1 |
| Brownian Motion | p. 4 |
| Fractional Brownian Motion | p. 5 |
| Stable Levy Processes | p. 9 |
| Lamperti Transformation | p. 11 |
| Some Historical Background | p. 13 |
| Fundamental Limit Theorem | p. 13 |
| Fixed Points of Renormalization Groups | p. 15 |
| Limit Theorems (I) | p. 16 |
| Selfsimilar Processes with Stationary Increments | p. 19 |
| Simple Properties | p. 19 |
| Long-Range Dependence (I) | p. 21 |
| Selfsimilar Processes with Finite Variances | p. 22 |
| Limit Theorems (II) | p. 24 |
| Stable Processes | p. 27 |
| Selfsimilar Processes with Infinite Variance | p. 29 |
| Long-Range Dependence (II) | p. 34 |
| Limit Theorems (III) | p. 37 |
| Fractional Brownian Motion | p. 43 |
| Sample Path Properties | p. 43 |
| Fractional Brownian Motion for H = 1/2 is not a Semimartingale | p. 45 |
| Stochastic Integrals with respect to Fractional Brownian Motion | p. 47 |
| Selected Topics on Fractional Brownian Motion | p. 51 |
| Distribution of the Maximum of Fractional Brownian Motion | p. 51 |
| Occupation Time of Fractional Brownian Motionv52 | |
| Multiple Points of Trajectories of Fractional Brownian Motion | p. 53 |
| Large Increments of Fractional Brownian Motion | p. 54 |
| Selfsimilar Processes with Independent Increments | p. 57 |
| K. Sato's Theorem | p. 57 |
| Getoor's Example | p. 60 |
| Kawazu's Example | p. 61 |
| A Gaussian Selfsimilar Process with Independent Increments | p. 62 |
| Sample Path Properties of Selfsimilar Stable Processes with Stationary Increments | p. 63 |
| Classification | p. 63 |
| Local Time and Nowhere Differentiability | p. 64 |
| Simulation of Selfsimilar Processes | p. 67 |
| Some References | p. 67 |
| Simulation of Stochastic Processes | p. 67 |
| Simulating Levy Jump Processes | p. 69 |
| Simulating Fractional Brownian Motion | p. 71 |
| Simulating General Selfsimilar Processes | p. 77 |
| Statistical Estimation | p. 81 |
| Heuristic Approaches | p. 81 |
| The R/S-Statistic | p. 82 |
| The Correlogram | p. 85 |
| Least Squares Regression in the Spectral Domain | p. 87 |
| Maximum Likelihood Methods | p. 87 |
| Further Techniques | p. 90 |
| Extensions | p. 93 |
| Operator Selfsimilar Processes | p. 93 |
| Semi-Selfsimilar Processes | p. 95 |
| References | p. 101 |
| Index | p. 109 |
| Table of Contents provided by Publisher. All Rights Reserved. |
ISBN: 9780691096278
ISBN-10: 0691096279
Series: Princeton Series in Applied Mathematics (Hardcover)
Audience:
Tertiary; University or College
Format:
Hardcover
Language:
English
Number Of Pages: 124
Published: 16th July 2002
Dimensions (cm): 24.2 x 16.1
x 1.6
Weight (kg): 0.35
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