## Stochastic Calculus for Fractional Brownian Motion and Related Processes, Issue 1929This volume examines the theory of fractional Brownian motion and other long-memory processes. Interesting topics for PhD students and specialists in probability theory, stochastic analysis and financial mathematics demonstrate the modern level of this field. It proves that the market with stock guided by the mixed model is arbitrage-free without any restriction on the dependence of the components and deduces different forms of the Black-Scholes equation for fractional market. |

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### Contents

11 The Elements of Fractional Calculus | 1 |

Definition and Elementary Properties | 7 |

13 Mandelbrotvan Ness Representation of fBm | 9 |

14 Fractional Brownian Motion with H ϵ¹₂1 on the White Noise Space | 10 |

15 Fractional Noise on White Noise Space | 12 |

16 Wiener Integration with Respect to fBm | 16 |

17 The Space of Gaussian Variables Generated by fBm | 24 |

18 Representation of fBm via the Wiener Process on a Finite Interval | 26 |

313 Some Other Results on Existence and Uniqueness of Solution of SDE Involving Processes Related to fBm with H 6 121 | 204 |

314 Some Properties of the Stochastic Differential Equations with Stationary Coefficients | 206 |

315 Semilinear Stochastic Differential Equations Involving Forward Integral wrt fBm | 220 |

316 Existence and Uniqueness of Solutions of SDE with TwoParameter Fractional Brownian Field | 223 |

32 The Mixed SDE Involving Both the Wiener Process and fBm | 225 |

322 The Existence and Uniqueness of the Solution of the Mixed SDE for fBm with H 341 | 227 |

323 The Girsanov Theorem and the Measure Transformation for the Mixed Semilinear SDE | 238 |

331 The Lipschitz and the Growth Conditions on the Negative Norms of Coefficients | 240 |

19 The Inequalities for the Moments of the Wiener Integrals with Respect to fBm | 35 |

110 Maximal Inequalities for the Moments of Wiener Integrals with Respect to fBm | 41 |

111 The Conditions of Continuity of Wiener Integrals with Respect to fBm | 54 |

112 The Estimates of Moments of the Solution of Simple Stochastic Differential Equations Involving fBm | 55 |

113 Stochastic Fubini Theorem for the Wiener Integrals wrt fBm | 57 |

114 Martingale Transforms and Girsanov Theorem for Longmemory Gaussian Processes | 58 |

How to Approximate Them by Semimartingales 1151 Approximation of fBm by Continuous Processes of Bounded Variation | 71 |

1152 Convergence Bᵸ¹BBᵸ in Besov Space W𝛌ab | 73 |

1153 Weak Convergence to fBm in the Schemes of Series | 78 |

116 Hölder Properties of the Trajectories of fBm and of Wiener Integrals wrt fBm | 87 |

117 Estimates for Fractional Derivatives of fBm and of Wiener Integrals wrt Wiener Process via the GarsiaRodemichRumsey Inequality | 88 |

118 Power Variations of fBm and of Wiener Integrals wrt fBm | 90 |

119 Lévy Theorem for fBm | 94 |

1201 The Main Definition 1202 Holder Properties of Twoparameter fBm | 117 |

1203 Fractional Integrals and Fractional Derivatives of Twoparameter Functions | 118 |

21 Pathwise Stochastic Integration 211 Pathwise Stochastic Integration in the Fractional Sobolevtype Spaces | 123 |

212 Pathwise Stochastic Integration in Fractional Besovtype Spaces | 128 |

221 Some Additional Properties of Twoparameter Fractional Integrals and Derivatives | 131 |

222 Generalized Twoparameter LebesgueStieltjes Integrals | 132 |

223 Generalized Integrals of Twoparameter fBm in the Case of the Integrand Depending on fBm | 136 |

225 The Existence of the Integrals of the Second Kind of a Twoparameter fBm | 137 |

H e 121 as 5integration 231 Wick Products and Sintegration | 141 |

232 Comparison of Wick and Pathwise Integrals for Markov Integrands | 145 |

233 Comparison of Wick and Stratonovich Integrals for General Integrands | 154 |

234 Reduction of Wick Integration wrt Fractional Noise to the Integration wrt White Noise | 157 |

24 Skorohod Forward Backward and Symmetric Integration wrt fBm Two Approaches to Skorohod Integration | 158 |

251 The Basic Idea | 162 |

252 First and Higherorder Integrals with Respect to X | 164 |

253 Generalized Integrals with Respect to fBm | 169 |

26 Stochastic Fubini Theorem for Stochastic Integrals wrt Fractional Brownian Motion | 174 |

271 The Simplest Version | 182 |

272 Ito Formula for Linear Combination of Fractional Brownian Motions with Hi 6 121 in Terms of Pathwise Integrals and Ito Integral | 183 |

273 The Ito Formula in Terms of Wick Integrals | 184 |

274 The Ito Formula for if 6 012 | 185 |

275 Ito Formula for Fractional Brownian Fields | 186 |

276 The Ito Formula for H 6 01 in Terms of Isometric Integrals and Its Applications | 189 |

281 The Girsanov Theorem for fBm | 191 |

282 When the Conditions of the Girsanov Theorem Are Fulfilled? Differentiability of the Fractional Integrals | 193 |

31 Stochastic Differential Equations Driven by Fractional Brownian Motion with Pathwise Integrals 311 Existence and Uniqueness of Solutions the R... | 197 |

312 Norm and Moment Estimates of Solution | 202 |

332 Quasilinear SDE with Fractional Noise | 241 |

34 The Rate of Convergence of Euler Approximations of Solutions of SDE Involving fBm | 243 |

341 Approximation of Pathwise Equations | 244 |

342 Approximation of Quasilinear Skorohodtype Equations | 255 |

35 Stochastic Differential Equation with Additive Wiener Integral wrt Fractional Noise | 262 |

351 Existence of a Weak Solution for Regular Coefficients | 263 |

352 Existence of a Weak Solution for SDE with Discontinuous Drift | 266 |

353 Uniqueness in Law and Pathwise Uniqueness for Regular Coefficients | 271 |

354 Existence of a Strong Solution for the Regular Case | 272 |

355 Existence of a Strong Solution for Discontinuous Drift | 274 |

356 Estimates of Moments of Solutions for Regular Case and H 6 012 | 278 |

357 The Estimates of the Norms of the Solution in the Orlicz Spaces | 280 |

358 The Distribution of the Supremum of the Process X on 0 T | 284 |

359 Modulus of Continuity of Solution of Equation Involving Fractional Brownian Motion | 287 |

41 Optimal Filtering of a Mixed BrownianFractionalBrownian Model with Fractional Brownian Observation Noise | 291 |

42 Optimal Filtering in Conditionally Gaussian Linear Systems with Mixed Signal and Fractional Brownian Observation Noise | 295 |

43 Optimal Filtering in Systems with Polynomial Fractional Brownian Noise | 298 |

Financial Applications of Fractional Brownian Motion 51 Discussion of the Arbitrage Problem 511 Longrange Dependence in Economics and Finance | 300 |

512 Arbitrage in Pure Fractional Brownian Model The Original Rogers Approach | 302 |

513 Arbitrage in the Pure Fractional Model Results of Shiryaev and Dasgupta | 304 |

Absence of Arbitrage and Related Topics | 305 |

515 Equilibrium of Financial Market The Fractional Burgers Equation | 321 |

521 The BlackScholes Equation for the Mixed BrownianFractionalBrownian Model | 322 |

and WickItˆoSkorohod Integral in the Problems of Arbitrage and Replication in the Fractional BlackScholes Pricing Model | 323 |

Statistical Inference with Fractional Brownian Motion 61 Testing Problems for the Density Process for fBm with Different Drifts | 327 |

611 Observations Based on the Whole Trajectory with a and H Known | 329 |

612 Discretely Observed Trajectory and a Unknown | 331 |

621 Introduction | 335 |

623 Goodnessoffit Tests with Discrete Observations | 337 |

624 On Volatility Estimation | 340 |

625 Goodnessoffit Test with Unknown fi and cr | 342 |

63 Parameter Estimates in the Models Involving fBm | 343 |

631 Consistency of the Drift Parameter Estimates in the Pure Fractional Brownian Diffusion Model | 344 |

632 Consistency of the Drift Parameter Estimates in the Mixed BrownianfractionalBrownian Diffusion Model with Linearly Dependent Wt and B1 | 349 |

633 The Properties of Maximum Likelihood Estimates in Diffusion BrownianFractionalBrownian Models with Independent Components | 354 |

Some Related Calculations | 363 |

Approximation of Beta Integrals and Estimation of Kernels | 365 |

References | 369 |

391 | |

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Stochastic Calculus for Fractional Brownian Motion and Related Processes Yuliya Mishura Limited preview - 2008 |

### Common terms and phrases

according to Theorem arbitrage assumptions Besov space Birkh¨auser Black–Scholes bounded calculus coefficients conditions of Theorem consider convergence Corollary defined Denote differential equals estimate Evidently fBm with Hurst fc=i fractional Brownian motion fractional integrals Fubini theorem Gaussian process Girsanov Theorem Holder continuous holds Hurst index inequality integral w.r.t. fBm Integrals with Respect interval involving fBm Itˆo Ito formula Lemma Let H G linear Long-Range Dependence martingale Math Mishura Moreover nonrandom norm Nualart obtain parameter pathwise integral Prob probability space proof follows properties prove random variable Remark representation Respect to fBm right-hand side satisfy Section semimartingale sequence similarly solution of equation Stoch stochastic diﬀerential equations Stochastic Integration stochastic process subsection Taqqu Theory tion trajectories two-parameter whence Wick products Wiener integrals Wiener integrals w.r.t. Wiener process