Stochastic Calculus for Fractional Brownian Motion and Related Processes, Issue 1929

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This 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 ϵ₂1 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 BᵸBBᵸ in Besov Space W𝛌ab
73
1153 Weak Convergence to fBm in the Schemes of Series
78
116 Hlder 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 Lvy 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
Index
391
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