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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.
List of contents
Preface.- Introduction.- Part I Examples of Self-Similar Processes.- 1.Fractional Brownian Motion and Related Processes.- 2.Solutions to the Linear Stochastic Heat and Wave Equation.- 3.Non Gaussian Self-Similar Processes.- 4.Multiparameter Gaussian Processes.- Part II Variations of Self-Similar Process: Central and Non-Central Limit Theorems.- 5.First and Second Order Quadratic Variations. Wavelet-Type Variations.- 6.Hermite Variations for Self-Similar Processes.- Appendices: A.Self-Similar Processes with Stationary Increments: Basic Properties.- B.Kolmogorov Continuity Theorem.- C.Multiple Wiener Integrals and Malliavin Derivatives.- References.- Index.
About the author
Ciprian Tudor is Full Professor at the University of Lille 1, France. He graduated from the University of Bucharest, Romania in 1998 and he obtained his PH.D. degree on Probability Theory from Université de La Rochelle, France in 2002. After the doctorate he worked at the Université Pierre et Marie Curie Paris 6, France and at the Université de Panthéon-Sorbonne Paris 1 where he obtained the Habilitation in 2006. He has published intensively on stochastic processes, especially Malliavin calculus, self-similar processes and their applications. Up to 2012 he has over 80 scientific publications in various international recognized journals on probability theory and statistics.
Summary
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.
Additional text
“The author provides the general theory for different classes of self-similar processes with a complete treatment of limit theorems for their variations. … The book is self-contained and suitable for both graduate students with a basic background in probability theory and stochastic processes and researchers whose aim is investigating this topic.” (Anthony Réveillac, Mathematical Reviews, February, 2015)
“This monograph is a profound survey of recent developments in the fields of … self-similar processes and their calculus of variations. … It may serve as an excellent basis for research seminars or special classes on Gaussian processes and Malliavin’s calculus and as a starting point for applied mathematicians with interest in self-similar processes.” (Michael Högele, zbMATH 1308.60004, 2015)
Report
"The author provides the general theory for different classes of self-similar processes with a complete treatment of limit theorems for their variations. ... The book is self-contained and suitable for both graduate students with a basic background in probability theory and stochastic processes and researchers whose aim is investigating this topic." (Anthony Réveillac, Mathematical Reviews, February, 2015)
"This monograph is a profound survey of recent developments in the fields of ... self-similar processes and their calculus of variations. ... It may serve as an excellent basis for research seminars or special classes on Gaussian processes and Malliavin's calculus and as a starting point for applied mathematicians with interest in self-similar processes." (Michael Högele, zbMATH 1308.60004, 2015)
