Stochastic Calculus for Fractional Brownian Motion and Applications

1. Front Matter

   Pages i-xiv

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2. Fractional Brownian motion

   1. Intrinsic properties of the fractional Brownian motion

      Pages 5-20

3. Stochastic calculus

   1. Wiener and divergence-type integrals for fractional Brownian motion

      Pages 23-45

   2. Fractional Wick Itô Skorohod (fWIS) integrals for fBm of Hurst index H >1/2

      Pages 47-97

   3. WickItô Skorohod (WIS) integrals for fractional Brownian motion

      Pages 99-122

   4. Pathwise integrals for fractional Brownian motion

      Pages 123-145

   5. A useful summary

      Pages 147-166

4. Applications of stochastic calculus

   1. Fractional Brownian motion in finance

      Pages 169-180

   2. Stochastic partial differential equations driven by fractional Brownian fields

      Pages 181-206

   3. Stochastic optimal control and applications

      Pages 207-238

   4. Local time for fractional Brownian motion

      Pages 239-269

5. Back Matter

   Pages 273-329

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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.

• Brownian motion
• Markov
• Markov process
• Martingale
• Potential
• Probability theory
• Semimartingale
• Stochastic calculus
• calculus
• equation
• fractional Brownian motion
• local time
• mathematics
• model
• partial differential equation

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)

• Mathematisches Institut, 80333, Germany

  Francesca Biagini

• University of Kansas, 66045-2142, Lawrence, USA

  Yaozhong Hu

• University of Oslo, N-0316, Blindern, Norway

  Bernt Øksendal

• University of Manchester, M13 9PL, UK

  Tusheng Zhang

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