Stochastic Calculus for Fractional Brownian Motion and Applications

by Biagini, Francesca; Hu, Yaozhong; Oksendal, Bernt; Zhang, Tusheng

• ISBN13: 9781852339968
• ISBN10: 1852339969
• Format: Hardcover
• Copyright: 2008-02-22
• Publisher: Springer Nature

Summary

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.

Table of Contents

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