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.

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. 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.

The first book to compare the different frameworks and methods of stochastic integration for fBm. It also discusses the applications of the resulting theory. Written by leading contributors to the field.

Autorentext

Massimo Campanino was born on 1952. On 1975 he got the degree on Mathematics at the University of Rome, Italy, under the supervisorship in Mathematics under the supervisorship of prof. Bruno de Finetti. Since 1990 he is full Professor of Probability at the University of Bologna, Italy. He has been visitor at I. H. E. S. of Bures-sur-Yvette (France), at the University of Princeton and at the University of California Irvine. He has performed research on random fields, dynamical systems random processes, systems with random potential and in random environment, percolation. He is the author of works in collaboration with H. Epstein and D. Ruelle where the existence of a solution of Feigenbaum's functional equation, related to the universal behaviour of one-dimensional dynamical systems, was first proved. In works in collaboration with D. Ioffe he proved the Ornstein-Zernike behaviour for Bernoulli percolation below the critical probability and then with D. Ioffe and Y. Velenik for finite range Ising models. He has been national coordinator of the 2006 Prin project "Percolation, random fields, evolution of stochastic interacting systems" and of the 2009 Prin project "Random fields, percolation and stochastic. Francesca Biagini was born in 1973. In Pisa she got the degree in Mathematics at the University in 1996 and attended the Scuola Normale, where she also obtained her PhD in Mathematics with specialization in Financial Mathematics in 2001. In 1999 she got a position as ricercatore at the University of Bologna. She moved then in 2005 to the University of Munich as associate professor. In 2009 she got the Chair of Financial and Insurance Mathematics at the University of Munich. She has been visiting at the University of Oslo, Evry, Toulose, Singapore, UCSB, Columbia University and Stockholm School of Economics. Her field of research concerns mainly martingale methods for financial and insurance markets and stochastic calculus for fractional Brownian motion.She is presently Member of the Council of the Bachelier Finance Society and associate editor for the journal Stochastic Analysis and Applications.

Inhalt
Fractional Brownian motion.- Intrinsic properties of the fractional Brownian motion.- Stochastic calculus.- Wiener and divergence-type integrals for fractional Brownian motion.- Fractional Wick Itô Skorohod (fWIS) integrals for fBm of Hurst index H >1/2.- WickItô Skorohod (WIS) integrals for fractional Brownian motion.- Pathwise integrals for fractional Brownian motion.- A useful summary.- 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.