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.
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Table of ContentsPreface.- 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.