Résumé
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.
Contents
Chapter 1. Introduction- Definition of Selfsimilarity
- Brownian Motion
- Fractional Brownian Motion
- Stable Lévy Processes
- Lamperti Transformation
- Fundamental Limit Theorem
- Fixed Points of Renormalization Groups
- Limit Theorems (I)
- Simple Properties
- Long-Range Dependence (I)
- Selfsimilar Processes with Finite Variances
- Limit Theorems (II)
- Stable Processes
- Selfsimilar Processes with Infinite Variance
- Long-Range Dependence (II)
- Limit Theorems (III)
- Sample Path Properties
- Fractional Brownian Motion for H = 1/2 is not a Semimartingale
- Stochastic Integrals with respect to Fractional Brownian Motion
- Selected Topics on Fractional Brownian Motion
- K. Sato's Theorem
- Getoor's Example
- Kawazu's Example
- A Gaussian Selfsimilar Process with Independent Increments
- Classification
- Local Time and Nowhere Differentiability
- Some References
- Simulation of Stochastic Processes
- Simulating Lévy Jump Processes
- Simulating Fractional Brownian Motion
- Simulating General Selfsimilar Processes
- Heuristic Approaches
- Maximum Likelihood Methods
- Further Techniques
- Operator Selfsimilar Processes
- Semi-Selfsimilar Processes
L'auteur - Paul Embrechts
Paul Embrechts, Professor of Insurance Mathematics at the Swiss Federal Institute of Technology (ETH) in Zurich, is the coauthor of Modelling Extremal Events for Insurance and Finance.
L'auteur - Makoto Maejima
Makoto Maejima is Professor of Mathematics at Keio
University, Yokohama, Japan. He has published extensively
on selfsimilarity and stable processes.
Caractéristiques techniques
| PAPIER | |
| Éditeur(s) | Princeton University Press |
| Auteur(s) | Paul Embrechts, Makoto Maejima |
| Parution | 09/12/2002 |
| Nb. de pages | 122 |
| Format | 16 x 24 |
| Couverture | Relié |
| Poids | 353g |
| Intérieur | Noir et Blanc |
| EAN13 | 9780691096278 |
| ISBN13 | 978-0-691-09627-8 |
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