Selfsimilar Processes

Paul Embrechts is Professor of Mathematics at the Swiss Federal Institute of Technology (ETHZ), Zurich, 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.

"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

• 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 Levy 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 Levy 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