Stochastic calculus for fractional Brownian motion and applications

Francesca Biagini, Yaozhong Hu, Bernt Øksendal, Tusheng Zhang (auth.)9781852339968, 1852339969, 9781846287978, 1846287979

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.

Table of contents :
Front Matter….Pages i-xiv
Intrinsic properties of the fractional Brownian motion….Pages 5-20
Wiener and divergence-type integrals for fractional Brownian motion….Pages 23-45
Fractional Wick Itô Skorohod (fWIS) integrals for fBm of Hurst index H >1/2….Pages 47-97
WickItô Skorohod (WIS) integrals for fractional Brownian motion….Pages 99-122
Pathwise integrals for fractional Brownian motion….Pages 123-145
A useful summary….Pages 147-166
Fractional Brownian motion in finance….Pages 169-180
Stochastic partial differential equations driven by fractional Brownian fields….Pages 181-206
Stochastic optimal control and applications….Pages 207-238
Local time for fractional Brownian motion….Pages 239-269
Back Matter….Pages 273-329