Selfsimilar Processes (Princeton Series in Applied Mathematics)

✍ Scribed by Paul Embrechts, Makoto Maejima

• Publisher: Princeton University Press
• Year: 2002
• Tongue: English
• Leaves: 124
• Series: Princeton Series in Applied Mathematics
• Category: Library

✦ Synopsis

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

Contents......Page 7
Preface......Page 9
1.1 Definition of Selfsimilarity......Page 11
1.2 Brownian Motion......Page 14
1.3 Fractional Brownian Motion......Page 15
1.4 Stable Lévy Processes......Page 19
1.5 Lamperti Transformation......Page 21
2.1 Fundamental Limit Theorem......Page 23
2.2 Fixed Points of Renormalization Groups......Page 25
2.3 Limit Theorems (I)......Page 26
3.1 Simple Properties......Page 29
3.2 Long-Range Dependence (I)......Page 31
3.3 Selfsimilar Processes with Finite Variances......Page 32
3.4 Limit Theorems (II)......Page 34
3.5 Stable Processes......Page 37
3.6 Selfsimilar Processes with Infinite Variance......Page 39
3.7 Long-Range Dependence (II)......Page 44
3.8 Limit Theorems (III)......Page 47
4.1 Sample Path Properties......Page 53
4.2 Fractional Brownian Motion for H ≠ 1/2 is not a Semimartingale......Page 55
4.3 Stochastic Integrals with respect to Fractional Brownian Motion......Page 57
4.4.1 Distribution of the Maximum of Fractional Brownian Motion......Page 61
4.4.2 Occupation Time of Fractional Brownian Motion......Page 62
4.4.3 Multiple Points of Trajectories of Fractional Brownian Motion......Page 63
4.4.4 Large Increments of Fractional Brownian Motion......Page 64
5.1 K. Sato’s Theorem......Page 67
5.2 Getoor’s Example......Page 70
5.3 Kawazu’s Example......Page 71
5.4 A Gaussian Selfsimilar Process with Independent Increments......Page 72
6.1 Classification......Page 73
6.2 Local Time and Nowhere Differentiability......Page 74
7.2 Simulation of Stochastic Processes......Page 77
7.3 Simulating Lévy Jump Processes......Page 79
7.4 Simulating Fractional Brownian Motion......Page 81
7.5 Simulating General Selfsimilar Processes......Page 87
8.1 Heuristic Approaches......Page 91
8.1.1 The R/S-Statistic......Page 92
8.1.2 The Correlogram......Page 95
8.2 Maximum Likelihood Methods......Page 97
8.3 Further Techniques......Page 100
9.1 Operator Selfsimilar Processes......Page 103
9.2 Semi-Selfsimilar Processes......Page 105
References......Page 111
L......Page 119
S......Page 120
W......Page 121