Selfsimilar Processes ：Selfsimilar Processes ( Princeton Series in Applied Mathematics )

Publication subTitle ：Selfsimilar Processes

Publication series ：Princeton Series in Applied Mathematics

Author： Embrechts Paul  

Publisher： Princeton University Press‎

Publication year： 2009

E-ISBN: 9781400825103

P-ISBN(Paperback): 9780691096278

Subject： O211.3 distribution theory

Keyword： 数理科学和化学

Language： ENG

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Description

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 mathemat

Chapter

Contents

Preface

Chapter 1. Introduction

1.1 Definition of Selfsimilarity

1.2 Brownian Motion

1.3 Fractional Brownian Motion

1.4 Stable Lévy Processes

1.5 Lamperti Transformation

Chapter 2. Some Historical Background

2.1 Fundamental Limit Theorem

2.2 Fixed Points of Renormalization Groups

2.3 Limit Theorems (I)

Chapter 3. Selfsimilar Processes with Stationary Increments

3.1 Simple Properties

3.2 Long-Range Dependence (I)

3.3 Selfsimilar Processes with Finite Variances

3.4 Limit Theorems (II)

3.5 Stable Processes

3.6 Selfsimilar Processes with Infinite Variance

3.7 Long-Range Dependence (II)

3.8 Limit Theorems (III)

Chapter 4. Fractional Brownian Motion

4.1 Sample Path Properties

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

4.3 Stochastic Integrals with respect to Fractional Brownian Motion

4.4 Selected Topics on Fractional Brownian Motion

4.4.1 Distribution of the Maximum of Fractional Brownian Motion

4.4.2 Occupation Time of Fractional Brownian Motion

4.4.3 Multiple Points of Trajectories of Fractional Brownian Motion

4.4.4 Large Increments of Fractional Brownian Motion

Chapter 5. Selfsimilar Processes with Independent Increments

5.1 K. Sato’s Theorem

5.2 Getoor’s Example

5.3 Kawazu’s Example

5.4 A Gaussian Selfsimilar Process with Independent Increments

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

6.1 Classification

6.2 Local Time and Nowhere Differentiability

Chapter 7. Simulation of Selfsimilar Processes

7.1 Some References

7.2 Simulation of Stochastic Processes

7.3 Simulating Lévy Jump Processes

7.4 Simulating Fractional Brownian Motion

7.5 Simulating General Selfsimilar Processes

Chapter 8. Statistical Estimation

8.1 Heuristic Approaches

8.1.1 The R/S-Statistic

8.1.2 The Correlogram

8.1.3 Least Squares Regression in the Spectral Domain

8.2 Maximum Likelihood Methods

8.3 Further Techniques

Chapter 9. Extensions

9.1 Operator Selfsimilar Processes

9.2 Semi-Selfsimilar Processes

References

Index

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