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Selfsimilar Processes : Princeton Series in Applied Mathematics - Paul Embrechts

Selfsimilar Processes

Princeton Series in Applied Mathematics

Hardcover

Published: 5th August 2002
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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.

"Authoritative and written by leading experts, this book is a significant contribution to a growing field. Selfsimilar processes crop up in a wide range of subjects from finance to physics, so this book will have a correspondingly wide readership."--Chris Rogers, Bath University
"This is a timely book. Everybody is talking about scaling, and selfsimilar stochastic processes are the basic and the clearest examples of models with scaling. In applications from finance to communication networks, selfsimilar processes are believed to be important. Yet much of what is known about them is folklore; this book fills the void and gives reader access to some hard facts. And because this book requires only modest mathematical sophistication, it is accessible to a wide audience."--Gennady Samorodnitsky, Cornell University

Prefacep. ix
Introductionp. 1
Definition of Selfsimilarityp. 1
Brownian Motionp. 4
Fractional Brownian Motionp. 5
Stable Levy Processesp. 9
Lamperti Transformationp. 11
Some Historical Backgroundp. 13
Fundamental Limit Theoremp. 13
Fixed Points of Renormalization Groupsp. 15
Limit Theorems (I)p. 16
Selfsimilar Processes with Stationary Incrementsp. 19
Simple Propertiesp. 19
Long-Range Dependence (I)p. 21
Selfsimilar Processes with Finite Variancesp. 22
Limit Theorems (II)p. 24
Stable Processesp. 27
Selfsimilar Processes with Infinite Variancep. 29
Long-Range Dependence (II)p. 34
Limit Theorems (III)p. 37
Fractional Brownian Motionp. 43
Sample Path Propertiesp. 43
Fractional Brownian Motion for H = 1/2 is not a Semimartingalep. 45
Stochastic Integrals with respect to Fractional Brownian Motionp. 47
Selected Topics on Fractional Brownian Motionp. 51
Distribution of the Maximum of Fractional Brownian Motionp. 51
Occupation Time of Fractional Brownian Motionv52
Multiple Points of Trajectories of Fractional Brownian Motionp. 53
Large Increments of Fractional Brownian Motionp. 54
Selfsimilar Processes with Independent Incrementsp. 57
K. Sato's Theoremp. 57
Getoor's Examplep. 60
Kawazu's Examplep. 61
A Gaussian Selfsimilar Process with Independent Incrementsp. 62
Sample Path Properties of Selfsimilar Stable Processes with Stationary Incrementsp. 63
Classificationp. 63
Local Time and Nowhere Differentiabilityp. 64
Simulation of Selfsimilar Processesp. 67
Some Referencesp. 67
Simulation of Stochastic Processesp. 67
Simulating Levy Jump Processesp. 69
Simulating Fractional Brownian Motionp. 71
Simulating General Selfsimilar Processesp. 77
Statistical Estimationp. 81
Heuristic Approachesp. 81
The R/S-Statisticp. 82
The Correlogramp. 85
Least Squares Regression in the Spectral Domainp. 87
Maximum Likelihood Methodsp. 87
Further Techniquesp. 90
Extensionsp. 93
Operator Selfsimilar Processesp. 93
Semi-Selfsimilar Processesp. 95
Referencesp. 101
Indexp. 109
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691096278
ISBN-10: 0691096279
Series: Princeton Series in Applied Mathematics
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 128
Published: 5th August 2002
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2  x 1.27
Weight (kg): 0.34