Selfsimilar Processes
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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.
Paul Embrechts
Paul Embrechts is Professor of Mathematics at the Swiss Federal Institute of Technology (ETHZ), Zürich, Switzerland. He is the author of numerous scientific papers on stochastic processes and their applications and the coauthor of the influential book on Modelling of Extremal Events for Insurance and Finance. Makoto Maejima is Professor of Mathematics at Keio University, Yokohama, Japan. He has published extensively on selfsimilarity and stable processes.
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Selfsimilar Processes - Paul Embrechts
Selfsimilar Processes
PRINCETON SERIES IN APPLIED MATHEMATICS
EDITORS
Daubechies, I. Princeton University
Weinan E. Princeton University
Lenstra, J.K. Technische Universiteit Eindhoven
Süli, E. University of Oxford
TITLES IN THE SERIES
Emil Simiu, Chaotic Transitions in Deterministic and Stochastic Dynamical Systems: Applications of Melnikov Processes in Engineering, Physics and Neuroscience
Paul Embrechts and Makoto Maejima, Selfsimilar Processes
Jiming Peng, Cornelis Roos and Tamás Terlaky, Self-Regularity: A New Paradigm for Primal-Dual Interior Point Algorithms
Selfsimilar Processes
Paul Embrechts and Makoto Maejima
PRINCETON UNIVERSITY PRESS
OXFORD, PRINCETON
Copyright © 2002 by Princeton University Press
Published by Princeton University Press,
41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
3 Market Place, Woodstock, Oxfordshire OX20 1SY
All Rights Reserved
Library of Congress Cataloging-in-Publication Data applied for.
Embrechts, Paul & Maejima, Makoto
Selfsimilar Processes/Paul Embrechts and Makoto Maejima
p.cm.
Includes bibliographical references and index.
ISBN: 978-1-40082-510-3
British Library Cataloging-in-Publication Data is available
This book has been composed in Times and Abadi
Printed on acid-free paper
www.pup.princeton.edu
Printed in the United States of America
Contents
Preface
Chapter 1. Introduction
1.1 Definition of Selfsimilarity
1.2 Brownian Motion
1.3 Fractional Brownian Motion
1.4 Stable Lévy Processes
1.5 Lamperti Transformation
Chapter 2. Some Historical Background
2.1 Fundamental Limit Theorem
2.2 Fixed Points of Renormalization Groups
2.3 Limit Theorems (I)
Chapter 3. Selfsimilar Processes with Stationary Increments
3.1 Simple Properties
3.2 Long-Range Dependence (I)
3.3 Selfsimilar Processes with Finite Variances
3.4 Limit Theorems (II)
3.5 Stable Processes
3.6 Selfsimilar Processes with Infinite Variance
3.7 Long-Range Dependence (II)
3.8 Limit Theorems (III)
Chapter 4. Fractional Brownian Motion
4.1 Sample Path Properties
4.2 Fractional Brownian Motion for H≠ 1/2 is not a Semimartingale
4.3 Stochastic Integrals with respect to Fractional Brownian Motion
4.4 Selected Topics on Fractional Brownian Motion
4.4.1 Distribution of the Maximum of Fractional Brownian Motion
4.4.2 Occupation Time of Fractional Brownian Motion
4.4.3 Multiple Points of Trajectories of Fractional Brownian Motion
4.4.4 Large Increments of Fractional Brownian Motion
Chapter 5. Selfsimilar Processes with Independent Increments
5.1 K. Sato’s Theorem
5.2 Getoor’s Example
5.3 Kawazu’s Example
5.4 A Gaussian Selfsimilar Process with Independent Increments
Chapter 6. Sample Path Properties of Selfsimilar Stable Processes with Stationary Increments
6.1 Classification
6.2 Local Time and Nowhere Differentiability
Chapter 7. Simulation of Selfsimilar Processes
7.1 Some References
7.2 Simulation of Stochastic Processes
7.3 Simulating Lévy Jump Processes
7.4 Simulating Fractional Brownian Motion
7.5 Simulating General Selfsimilar Processes
Chapter 8. Statistical Estimation
8.1 Heuristic Approaches
8.1.1 The R/S-Statistic
8.1.2 The Correlogram
8.1.3 Least Squares Regression in the Spectral Domain
8.2 Maximum Likelihood Methods
8.3 Further Techniques
Chapter 9. Extensions
9.1 Operator Selfsimilar Processes
9.2 Semi-Selfsimilar Processes
References
Preface
First, a word about the title ‘‘Selfsimilar Processes’’. Let