Synopses & Reviews
Fractional Brownian motion (fBm) has been widely used to model a number of phenomena in diverse fields from biology to finance. This huge range of potential applications makes fBm an interesting object of study. fBm represents a natural one-parameter extension of classical Brownian motion therefore it is natural to ask if a stochastic calculus for fBm can be developed. This is not obvious, since fBm is neither a semimartingale (except when H = ½), nor a Markov process so the classical mathematical machineries for stochastic calculus are not available in the fBm case. Several approaches have been used to develop the concept of stochastic calculus for fBm. The purpose of this book is to present a comprehensive account of the different definitions of stochastic integration for fBm, and to give applications of the resulting theory. Particular emphasis is placed on studying the relations between the different approaches. Readers are assumed to be familiar with probability theory and stochastic analysis, although the mathematical techniques used in the book are thoroughly exposed and some of the necessary prerequisites, such as classical white noise theory and fractional calculus, are recalled in the appendices. This book will be a valuable reference for graduate students and researchers in mathematics, biology, meteorology, physics, engineering and finance. Aspects of the book will also be useful in other fields where fBm can be used as a model for applications.
Review
From the reviews: "The development of stochastic integration with respect to fBm continues to be a very active area of research ... became a necessity to collect the different approaches into a single monograph, in order to allow researchers in this field to have a general and quick view of the state of the art. This book very nicely attains this aim, and I can recommend it to any person interested in fractional Brownian motion." (Ivan Nourdin, Mathematical Reviews, Issue 2010 a)
Synopsis
Classical Brownian Motion is the fundamental mathematical tool used in modelling continuous time stochastic processes ??? they are used in physics, engineering, finance, and many other areas. In recent years it has been observed that fraction Brownian motion (fBm) possesses several properties which suggest its use in finance and other areas. However, the classical stochastic calculus does not apply to fBm, and the authors have been leading contributors to a theory of integration for fBe, which uses a white noise approach.
This book will discuss fBm from this stochastic calculus point of view, and develops many important results for its application ??? it will be the first monograph on this subject. This is a new and hotly debated method, and researchers will be very interested to see what the authors are proposing.
Synopsis
The purpose of this book is to present a comprehensive account of the different definitions of stochastic integration for fBm, and to give applications of the resulting theory. Particular emphasis is placed on studying the relations between the different approaches.
Table of Contents
Part I: Fractional Brownian Motion.- Intrinsic properties of the fractional Brownian motion. Part II: Stochastic Calculus.- Wiener and divergence-type integrals for fractional Brownian motion.- Fractional Wick-Itô-Skorohod (fWIS-) integrals for fractional Brownian motion of Hurst Index H > ½.- Wick-Itô-Skorohod integrals for fractional Brownian motion.- Pathwise integrals for fractional Brownian motion.- A useful summary. Part III: Applications of Stochastic Calculus.- Fractional Brownian motion in finance.- Stochastic partial differential equations driven by fractional Brownian fields.- Stochastic optimal control and applications.- Local time for fractional Brownian motion. Part IV: Appendices.- Classical Malliavin calculus.- Notions from fractional calculus.- Estimation of Hurst parameter.- Stochastic differential equations for fBm.- References.- List of symbols and notation.- Index.