|
Endorsements: "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 |
|
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 | 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 | ||