"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

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.

Preface ix

Chapter 1. Introduction 1

1.1 Definition of Selfsimilarity 1

1.2 Brownian Motion 4

1.3 Fractional Brownian Motion 5

1.4 Stable Lévy Processes 9

1.5 Lamperti Transformation 11

Chapter 2. Some Historical Background 13

2.1 Fundamental Limit Theorem 13

2.2 Fixed Points of Renormalization Groups 15

2.3 Limit Theorems (I) 16

Chapter 3. Selfsimilar Processes with Stationary Increments 19

3.1 Simple Properties 19

3.2 Long-Range Dependence (I) 21

3.3 Selfsimilar Processes with Finite Variances 22

3.4 Limit Theorems (II) 24

3.5 Stable Processes 27

3.6 Selfsimilar Processes with Infinite Variance 29

3.7 Long-Range Dependence (II) 34

3.8 Limit Theorems (III) 37

Chapter 4. Fractional Brownian Motion 43

4.1 Sample Path Properties 43

4.2 Fractional Brownian Motion for H = 1/2 is not a Semimartingale 45

4.3 Stochastic Integrals with respect to Fractional Brownian Motion 47

4.4 Selected Topics on Fractional Brownian Motion 51

4.4.1 Distribution of the Maximum of Fractional Brownian Motion 51

4.4.2 Occupation Time of Fractional Brownian Motion 52

4.4.3 Multiple Points of Trajectories of Fractional Brownian Motion 53

4.4.4 Large Increments of Fractional Brownian Motion 54.Chapter 5. Selfsimilar Processes with Independent Increments 57

5.1 K. Sato's Theorem 57

5.2 Getoor's Example 60

5.3 Kawazu's Example 61

5.4 A Gaussian Selfsimilar Process with Independent Increments 62

Chapter 6. Sample Path Properties of Selfsimilar Stable Processes with Stationary Increments 63

6.1 Classification 63

6.2 Local Time and Nowhere Differentiability 64

Chapter 7. Simulation of Selfsimilar Processes 67

7.1 Some References 67

7.2 Simulation of Stochastic Processes 67

7.3 Simulating Lévy Jump Processes 69

7.4 Simulating Fractional Brownian Motion 71

7.5 Simulating General Selfsimilar Processes 77

Chapter 8. Statistical Estimation 81

8.1 Heuristic Approaches 81

8.1.1 The R/S-Statistic 82

8.1.2 The Correlogram 85

8.1.3 Least Squares Regression in the Spectral Domain 87

8.2 Maximum Likelihood Methods 87

8.3 Further Techniques 90

Chapter 9. Extensions 93

9.1 Operator Selfsimilar Processes 93

9.2 Semi-Selfsimilar Processes 95

References 101

Index 109