| 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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