Analysis of Variations for Self-similar Processes

A Stochastic Calculus Approach

verfasst von: Ciprian Tudor

Verlag: Springer International Publishing

Buchreihe : Probability and its Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft" , Springer Professional "Technik"

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Über dieses Buch

Self-similar processes are stochastic processes that are invariant in distribution under suitable time scaling, and are a subject intensively studied in the last few decades. This book presents the basic properties of these processes and focuses on the study of their variation using stochastic analysis. While self-similar processes, and especially fractional Brownian motion, have been discussed in several books, some new classes have recently emerged in the scientific literature. Some of them are extensions of fractional Brownian motion (bifractional Brownian motion, subtractional Brownian motion, Hermite processes), while others are solutions to the partial differential equations driven by fractional noises.

In this monograph the author discusses the basic properties of these new classes of self-similar processes and their interrelationship. At the same time a new approach (based on stochastic calculus, especially Malliavin calculus) to studying the behavior of the variations of self-similar processes has been developed over the last decade. This work surveys these recent techniques and findings on limit theorems and Malliavin calculus.

Inhaltsverzeichnis

Frontmatter

Examples of Self-similar Processes

Frontmatter

Chapter 1. Fractional Brownian Motion and Related Processes

Abstract

Chapter 1 surveys the basic properties of fractional Brownian motion and related processes. Fractional Brownian motion is the only Gaussian self-similar process with stationary increments. Its applications to various area are now widely recognized. Recently, other Gaussian self-similar processes, connected with the fractional Brownian motion (the bifractional Brownian motion, the subfractional Brownian motion etc.), have been the object of the study in the scientific literature. We discuss the properties of these processes, including the regularity of their sample paths, the stochastic integral representation, the long-range dependence or the existence of their quadratic variations. We also analyze their interconnections.

Ciprian A. Tudor

Chapter 2. Solutions to the Linear Stochastic Heat and Wave Equation

Abstract

The solutions to certain stochastic partial differential equations with linear Gaussian noise constitute interesting examples of self-similar processes. In this chapter we analyze these classes of self-similar processes. We focus on the solution to the linear heat and wave equation driven by a Gaussian noise which behaves as a Brownian motion or fractional Brownian motion with respect to the time variable and is white or colored with respect to the space variable. We consider various aspects of these self-similar processes. In particular we present the conditions for the existence of the solution, the sharp regularity of their trajectories, we study the law of the solution to the linear heat equation and its connection with the bifractional Brownian motion.

Ciprian A. Tudor

Chapter 3. Non-Gaussian Self-similar Processes

Abstract

Chapter 3 is consecrated to the study of the Hermite processes. The Hermite processes are self-similar processes with stationary increments. They appear as limits in the so called Non-Central Limit Theorem. This class includes the fractional Brownian motion but all the other processes in the class of Hermite processes are non-Gaussian. Another interesting example in this class is the Rosenblatt process, which is discussed is details, together with its variants, in this part of the monograph.

Ciprian A. Tudor

Chapter 4. Multiparameter Gaussian Processes

Abstract

Chapter 4 treats multiparameter self-similar processes. The main focus is on the fractional Brownian sheet and on the multiparameter Hermite processes. We describe various properties of these processes such as the self-similarity, the stationarity of the increments, continuity or their stochastic integral representation.

Ciprian A. Tudor

Variations of Self-similar Processes: Central and Non-Central Limit Theorems

Frontmatter

Chapter 5. First and Second Order Quadratic Variations. Wavelet-Type Variations

Abstract

In this chapter we study the asymptotic behavior of the quadratic variation (including standard quadratic, quadratic variation based on higher order increments or wavelet-type quadratic variations) for several self-similar processes, such as fractional Brownian motion, the Rosenblatt process, the Hermite process of general order or the solution to the linear heat equation. We prove Central or Non-Central Limit Theorems for the sequence of quadratic variations using chaos expansion into multiple Wiener-Itô integrals and Malliavin calculus.

Ciprian A. Tudor

Chapter 6. Hermite Variations for Self-similar Processes

Abstract

Hermite variations are variants of the p-variations of stochastic processes, involving the Hermite polynomials applied to the increment of the process under analysis. This type of variations fits well into Malliavin calculus context. We discuss in this chapter the limit behavior in distribution of the Hermite variations of the fractional Brownian motion, fractional Brownian sheet and moving—average sequences. The chapter also presents Hsu-Robbins and Spitzer theorems corresponding to the limit behavior in distribution of the Hermite variations.

Ciprian A. Tudor

Backmatter

Titel: Analysis of Variations for Self-similar Processes
verfasst von: Ciprian Tudor
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-00936-0
Print ISBN: 978-3-319-00935-3
DOI: https://doi.org/10.1007/978-3-319-00936-0