Analysis Of Variations For Self Similar Processes

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.

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.

The articles in this collection are a sampling of some of the research presented during the conference “Stochastic Analysis and Related Topics”, held in May of 2015 at Purdue University in honor of the 60th birthday of Rodrigo Bañuelos. A wide variety of topics in probability theory is covered in these proceedings, including heat kernel estimates, Malliavin calculus, rough paths differential equations, Lévy processes, Brownian motion on manifolds, and spin glasses, among other topics.

A modern and rigorous introduction to long-range dependence and self-similarity, complemented by numerous more specialized up-to-date topics in this research area.

This book offers an introduction to the field of stochastic analysis of Hermite processes. These selfsimilar stochastic processes with stationary increments live in a Wiener chaos and include the fractional Brownian motion, the only Gaussian process in this class. Using the Wiener chaos theory and multiple stochastic integrals, the book covers the main properties of Hermite processes and their multiparameter counterparts, the Hermite sheets. It delves into the probability distribution of these stochastic processes and their sample paths, while also presenting the basics of stochastic integration theory with respect to Hermite processes and sheets. The book goes beyond theory and provides a thorough analysis of physical models driven by Hermite noise, including the Hermite Ornstein-Uhlenbeck process and the solution to the stochastic heat equation driven by such a random perturbation. Moreover, it explores up-to-date topics central to current research in statistical inference for Hermite-driven models.

This book develops alternative methods to estimate the unknown parameters in stochastic volatility models, offering a new approach to test model accuracy. While there is ample research to document stochastic differential equation models driven by Brownian motion based on discrete observations of the underlying diffusion process, these traditional methods often fail to estimate the unknown parameters in the unobserved volatility processes. This text studies the second order rate of weak convergence to normality to obtain refined inference results like confidence interval, as well as nontraditional continuous time stochastic volatility models driven by fractional Levy processes. By incorporating jumps and long memory into the volatility process, these new methods will help better predict option pricing and stock market crash risk. Some simulation algorithms for numerical experiments are provided.

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.