Selfsimilar Processes

by Embrechts, Paul

• ISBN13: 9780691096278
• ISBN10: 0691096279
• Format: Hardcover
• Copyright: 2002-07-16
• Publisher: Princeton Univ Pr

Summary

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.

Table of Contents

Preface	ix
Introduction	1	(12)
Definition of Selfsimilarity	1	(3)
Brownian Motion	4	(1)
Fractional Brownian Motion	5	(4)
Stable Levy Processes	9	(2)
Lamperti Transformation	11	(2)
Some Historical Background	13	(6)
Fundamental Limit Theorem	13	(2)
Fixed Points of Renormalization Groups	15	(1)
Limit Theorems (I)	16	(3)
Selfsimilar Processes with Stationary Increments	19	(24)
Simple Properties	19	(2)
Long-Range Dependence (I)	21	(1)
Selfsimilar Processes with Finite Variances	22	(2)
Limit Theorems (II)	24	(3)
Stable Processes	27	(2)
Selfsimilar Processes with Infinite Variance	29	(5)
Long-Range Dependence (II)	34	(3)
Limit Theorems (III)	37	(6)
Fractional Brownian Motion	43	(14)
Sample Path Properties	43	(2)
Fractional Brownian Motion for H ≠ 1/2 is not a Semimartingale	45	(2)
Stochastic Integrals with respect to Fractional Brownian Motion	47	(4)
Selected Topics on Fractional Brownian Motion	51	(6)
Distribution of the Maximum of Fractional Brownian Motion	51	(1)
Occupation Time of Fractional Brownian Motion	52	(1)
Multiple Points of Trajectories of Fractional Brownian Motion	53	(1)
Large Increments of Fractional Brownian Motion	54	(3)
Selfsimilar Processes with Independent Increments	57	(6)
K. Sato's Theorem	57	(3)
Getoor's Example	60	(1)
Kawazu's Example	61	(1)
A Gaussian Selfsimilar Process with Independent Increments	62	(1)
Sample Path Properties of Selfsimilar Stable Processes with Stationary Increments	63	(4)
Classification	63	(1)
Local Time and Nowhere Differentiability	64	(3)
Simulation of Selfsimilar Processes	67	(14)
Some References	67	(1)
Simulation of Stochastic Processes	67	(2)
Simulating Levy lump Processes	69	(2)
Simulating Fractional Brownian Motion	71	(6)
Simulating General Selfsimilar Processes	77	(4)
Statistical Estimation	81	(12)
Heuristic Approaches	81	(6)
The R/S-Statistic	82	(3)
The Correlogram	85	(2)
Least Squares Regression in the Spectral Domain	87	(1)
Maximum Likelihood Methods	87	(3)
Further Techniques	90	(3)
Extensions	93	(8)
Operator Selfsimilar Processes	93	(2)
Semi-Selfsimilar Processes	95	(6)
References	101	(8)
Index	109