| Wiener Integration with Respect to Fractional Brownian Motion | p. 1 |
| The Elements of Fractional Calculus | p. 1 |
| Fractional Brownian Motion: Definition and Elementary Properties | p. 7 |
| Mandelbrot-van Ness Representation of fBm | p. 9 |
| Fractional Brownian Motion with H ¿ (1/2, 1) on the White Noise Space | p. 10 |
| Fractional Noise on White Noise Space | p. 12 |
| Wiener Integration with Respect to fBm | p. 16 |
| The Space of Gaussian Variables Generated by fBm | p. 24 |
| Representation of fBm via the Wiener Process on a Finite Interval | p. 26 |
| The Inequalities for the Moments of the Wiener Integrals with Respect to fBm | p. 35 |
| Maximal Inequalities for the Moments of Wiener Integrals with Respect to fBm | p. 41 |
| The Conditions of Continuity of Wiener Integrals with Respect to fBm | p. 54 |
| The Estimates of Moments of the Solution of Simple Stochastic Differential Equations Involving fBm | p. 55 |
| Stochastic Fubini Theorem for the Wiener Integrals w.r.t fBm | p. 57 |
| Martingale Transforms and Girsanov Theorem for Long-memory Gaussian Processes | p. 58 |
| Nonsemimartingale Properties of fBm; How to Approximate Them by Semimartingales | p. 71 |
| Approximation of fBm by Continuous Processes of Bounded Variation | p. 71 |
| Convergence B 2+H,ß → B2+H in Besov Space W2+¿ [a,b] | p. 73 |
| Weak Convergence to fBm in the Schemes of Series | p. 78 |
| Holder Properties of the Trajectories of fBm and of Wiener Integrals w.r.t. fBm | p. 87 |
| Estimates for Fractional Derivatives of fBm and of Wiener Integrals w.r.t. Wiener Process via the Garsia-Rodemich-Rumsey Inequality | p. 88 |
| Power Variations of fBm and of Wiener Integrals w.r.t fBm | p. 90 |
| Levy Theorem for fBm | p. 94 |
| Multi-parameter Fractional Brownian Motion | p. 117 |
| The Main Definition | p. 117 |
| Holder Properties of Two-parameter fBm | p. 117 |
| Fractional Integrals and Fractional Derivatives of Two-parameter Functions | p. 118 |
| Stochastic Integration with Respect to fBm and Related Topics | p. 123 |
| Pathwise Stochastic Integration | p. 123 |
| Pathwise Stochastic Integration in the Fractional Sobolev-type Spaces | p. 123 |
| Pathwise Stochastic Integration in Fractional Besov-type Spaces | p. 128 |
| Pathwise Stochastic Integration w.r.t Multi-parameter fBm | p. 131 |
| Some Additional Properties of Two-parameter Fractional Integrals and Derivatives | p. 131 |
| Generalized Two-parameter Lebesgue-Stieltjes Integrals | p. 132 |
| Generalized Integrals of Two-parameter fBm in the Case of the Integrand Depending on fBm | p. 136 |
| Pathwise Integration in Two-parameter Besov Spaces | p. 136 |
| The Existence of the Integrals of the Second Kind of a Two-parameter fBm | p. 137 |
| Wick Integration with Respect to fBm with H ¿ [1/2,1) as S* -integration | p. 141 |
| Wick Products and S* -integration | p. 141 |
| Comparison of Wick and Pathwise Integrals for "Markov" Integrands | p. 145 |
| Comparison of Wick and Stratonovich Integrals for "General" Integrands | p. 154 |
| Reduction of Wick Integration w.r.t. Fractional Noise to the Integration w.r.t. White Noise | p. 157 |
| Skorohod, Forward, Backward and Symmetric Integration w.r.t. fBm. Two Approaches to Skorohod Integration | p. 158 |
| Isometric Approach to Stochastic Integration with Respect to fBm | p. 162 |
| The Basic Idea | p. 162 |
| First- and Higher-order Integrals with Respect to X | p. 164 |
| Generalized Integrals with Respect to fBm | p. 169 |
| Stochastic Fubini Theorem for Stochastic Integrals w.r.t. Fractional Brownian Motion | p. 174 |
| The Ito Formula for Fractional Brownian Motion | p. 182 |
| The Simplest Version | p. 182 |
| Ito Formula for Linear Combination of Fractional Brownian Motions with Hi ¿ [1/2,1] in Terms of Pathwise Integrals and Itô Integral | p. 183 |
| The Itô Formula in Terms of Wick Integrals | p. 184 |
| The Itô Formula for H ¿ (0,1/2) | p. 185 |
| Itô Formula for Fractional Brownian Fields | p. 186 |
| The Itô Formula for H ¿ (0,1) in Terms of Isometric Integrals, and Its Applications | p. 189 |
| The Girsanov Theorem for fBm and Its Applications | p. 191 |
| The Girsanov Theorem for fBm | p. 191 |
| When the Conditions of the Girsanov Theorem Are Fulfilled? Differentiability of the Fractional Integrals | p. 193 |
| Stochastic Differential Equations Involving Fractional Brownian Motion | p. 197 |
| Stochastic Differential Equations Driven by Fractional Brownian Motion with Pathwise Integrals | p. 197 |
| Existence and Uniqueness of Solutions: the Results of Nualart and R&acaron;şcanu | p. 197 |
| Norm and Moment Estimates of Solution | p. 202 |
| Some Other Results on Existence and Uniqueness of Solution of SDE Involving Processes Related to fBm with (H ¿ (1/2,1)) | p. 204 |
| Some Properties of the Stochastic Differential Equations with Stationary Coefficients | p. 206 |
| Semilinear Stochastic Differential Equations Involving Forward Integral w.r.t. fBm | p. 220 |
| Existence and Uniqueness of Solutions of SDE with Two-Parameter Fractional Brownian Field | p. 223 |
| The Mixed SDE Involving Both the Wiener Process and fBm | p. 225 |
| The Existence and Uniqueness of the Solution of the Mixed Semilinear SDE | p. 225 |
| The Existence and Uniqueness of the Solution of the Mixed SDE for fBm with H ¿ (3/4,1) | p. 227 |
| The Girsanov Theorem and the Measure Transformation for the Mixed Semilinear SDE | p. 238 |
| Stochastic Differential Equations with Fractional White Noise | p. 240 |
| The Lipschitz and the Growth Conditions on the Negative Norms of Coefficients | p. 240 |
| Quasilinear SDE with Fractional Noise | p. 241 |
| The Rate of Convergence of Euler Approximations of Solutions of SDE Involving fBm | p. 243 |
| Approximation of Pathwise Equations | p. 244 |
| Approximation of Quasilinear Skorohod-type Equations | p. 255 |
| SDE with the Additive Wiener Integral w.r.t. Fractional Noise | p. 262 |
| Existence of a Weak Solution for Regular Coefficients | p. 263 |
| Existence of a Weak Solution for SDE with Discontinuous Drift | p. 266 |
| Uniqueness in Law and Pathwise Uniqueness for Regular Coefficients | p. 271 |
| Existence of a Strong solution for the Regular Case | p. 272 |
| Existence of a Strong Solution for Discontinuous Drift | p. 274 |
| Estimates of Moments of Solutions for Regular Case and H ¿ (0,1/2) | p. 278 |
| The Estimates of the Norms of the Solution in the Orlicz Spaces | p. 280 |
| The Distribution of the Supremum of the Process X on [0,T] | p. 284 |
| Modulus of Continuity of Solution of Equation Involving Fractional Brownian Motion | p. 287 |
| Filtering in Systems with Fractional Brownian Noise | p. 291 |
| Optimal Filtering of a Mixed Brownian-Fractional-Brownian Model with Fractional Brownian Observation Noise | p. 291 |
| Optimal Filtering in Conditionally Gaussian Linear Systems with Mixed Signal and Fractional Brownian Observation Noise | p. 295 |
| Optimal Filtering in Systems with Polynomial Fractional Brownian Noise | p. 298 |
| Financial Applications of Fractional Brownian Motion | p. 301 |
| Discussion of the Arbitrage Problem | p. 301 |
| Long-range Dependence in Economics and Finance | p. 301 |
| Arbitrage in "Pure" Fractional Brownian Model. The Original Rogers Approach | p. 302 |
| Arbitrage in the "Pure" Fractional Model. Results of Shiryaev and Dasgupta | p. 304 |
| Mixed Brownian-Fractional-Brownian Model: Absence of Arbitrage and Related Topics | p. 305 |
| Equilibrium of Financial Market. The Fractional Burgers Equation | p. 321 |
| The Different Forms of the Black-Scholes Equation | p. 322 |
| The Black-Scholes Equation for the Mixed Brownian-Fractional-Brownian Model | p. 322 |
| Discussion of the Place of Wick Products and Wick-Ito-Skorohod Integral in the Problems of Arbitrage and Replication in the Fractional Black-Scholes Pricing Model | p. 323 |
| Statistical Inference with Fractional Brownian Motion | p. 327 |
| Testing Problems for the Density Process for fBm with Different Drifts | p. 327 |
| Observations Based on the Whole Trajectory with ¿ and H Known | p. 329 |
| Discretely Observed Trajectory and ¿ Unknown | p. 331 |
| Goodness-of-fit Test | p. 335 |
| Introduction | p. 335 |
| The Whole Trajectory Is Observed and the Parameters ¿ and ¿ Are Known | p. 335 |
| Goodness-of-fit Tests with Discrete Observations | p. 337 |
| On Volatility Estimation | p. 340 |
| Goodness-of-fit Test with Unknown ¿ and ¿ | p. 342 |
| Parameter Estimates in the Models Involving fBm | p. 343 |
| Consistency of the Drift Parameter Estimates in the Pure Fractional Brownian Diffusion Model | p. 344 |
| Consistency of the Drift Parameter Estimates in the Mixed Brownian-fractional-Brownian Diffusion Model with "Linearly" Dependent Wt and B2+Ht | p. 349 |
| The Properties of Maximum Likelihood Estimates in Diffusion Brownian-Fractional-Brownian Models with Independent Components | p. 354 |
| Mandelbrot-van Ness Representation: Some Related Calculations | p. 363 |
| Approximation of Beta Integrals and Estimation of Kernels | p. 365 |
| References | p. 369 |
| Index | p. 391 |
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