Stochastic Calculus for Fractional Brownian Motion and Applications / Edition 1

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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.

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 = 1/2), 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.

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Editorial Reviews

From the Publisher
From the reviews:

“The development of shastic 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)

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Product Details

  • ISBN-13: 9781852339968
  • Publisher: Springer London
  • Publication date: 2/28/2008
  • Series: Probability and Its Applications Series
  • Edition description: 2008
  • Edition number: 1
  • Pages: 330
  • Product dimensions: 0.81 (w) x 9.21 (h) x 6.14 (d)

Table of Contents

Preface     VII
Introduction     1
Fractional Brownian motion
Intrinsic properties of the fractional Brownian motion     5
Fractional Brownian motion     5
Stochastic integral representation     6
Correlation between two increments     8
Long-range dependence     9
Self-similarity     10
Holder continuity     11
Path differentiability     11
The fBm is not a semimartingale for H [not equal] 1/2     12
Invariance principle     14
Stochastic calculus
Wiener and divergence-type integrals for fractional Brownian motion     23
Wiener integrals     23
Wiener integrals for H > 1/2     27
Wiener integrals for H < 1/2     34
Divergence-type integrals for fBm     37
Divergence-type integral for H > 1/2     39
Divergence-type integral for H < 1/2     41
Fractional Wick Ito Skorohod (fWIS) integrals for fBm of Hurst index H > 1/2     47
Fractional white noise     47
Fractional Girsanov theorem     59
Fractional stochastic gradient     62
Fractional Wick Ito Skorohod integral     64
The [phi]-derivative     65
Fractional Wick Ito Skorohod integrals in L[superscript 2]     68
An Ito formula     71
L[superscript p] estimate for the fWIS integral     75
Iterated integrals and chaos expansion     78
Fractional Clark Hausmann Ocone theorem     83
Multidimensional fWIS integral     87
Relation between the fWIS integral and the divergence-type integral for H > 1/2     96
Wick Ito Skorohod (WIS) integrals for fractional Brownian motion     99
The M operator     99
The Wick Ito Skorohod (WIS) integral     103
Girsanov theorem     109
Differentiation     110
Relation with the standard Malliavin calculus     115
The multidimensional case     118
Pathwise integrals for fractional Brownian motion     123
Symmetric, forward and backward integrals for fBm     123
On the link between fractional and stochastic calculus     125
The case H < 1/2     126
Relation with the divergence integral     130
Relation with the fWIS integral     132
Relation with the WIS integral     137
A useful summary     147
Integrals with respect to fBm     147
Wiener integrals      147
Divergence-type integrals     150
fWIS integrals     151
WIS integrals     153
Pathwise integrals     154
Relations among the different definitions of stochastic integral     155
Relation between Wiener integrals and the divergence     156
Relation between the divergence and the fWIS integral     156
Relation between the fWIS and the WIS integrals     157
Relations with the pathwise integrals     158
Ito formulas with respect to fBm     160
Applications of stochastic calculus
Fractional Brownian motion in finance     169
The pathwise integration model (1/2 < H < 1)     170
The WIS integration model (0 < H < 1)     172
A connection between the pathwise and the WIS model     179
Concluding remarks     180
Stochastic partial differential equations driven by fractional Brownian fields     181
Fractional Brownian fields     181
Multiparameter fractional white noise calculus     185
The stochastic Poisson equation     189
The linear heat equation     194
The quasi-linear stochastic fractional heat equation     198
Stochastic optimal control and applications     207
Fractional backward stochastic differential equations     207
A stochastic maximum principle     211
Linear quadratic control     216
A minimal variance hedging problem     218
Optimal consumption and portfolio in a fractional Black and Scholes market     221
Optimal consumption and portfolio in presence of stochastic volatility driven by fBm     232
Local time for fractional Brownian motion     239
Local time for fBm     239
The chaos expansion of local time for fBm     245
Weighted local time for fBm     250
A Meyer Tanaka formula for fBm     253
A Meyer Tanaka formula for geometric fBm     255
Renormalized self-intersection local time for fBm     258
Application to finance     266
Classical Malliavin calculus     273
Classical white noise theory     273
Stochastic integration     278
Malliavin derivative     281
Notions from fractional calculus     285
Fractional calculus on an interval     285
Fractional calculus on the whole real line     288
Estimation of Hurst parameter     289
Absolute value method     290
Variance Method      290
Variance residuals methods     290
Hurst's rescaled range (R/S) analysis     291
Periodogram method     291
Discrete variation method     291
Whittle method     292
Maximum likelihood estimator     293
Quasi maximum likelihood estimator     294
Stochastic differential equations for fractional Brownian motion     297
Stochastic differential equations with Wiener integrals     297
Stochastic differential equations with pathwise integrals     300
Stochastic differential equations via rough path analysis     305
Rough path analysis     305
Stochastic calculus with rough path analysis     306
References     309
Index of symbols and notation     321
Index     325
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