Analysis of Variations for Self-similar Processes ― A Stochastic Calculus Approach

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.

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 Universite de La Rochelle, France in 2002. After the doctorate he worked at the Universite Pierre et Marie Curie Paris 6, France and at the Universite de Pantheon-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.