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by Oksana Banna, Yuliya Mishura, Kostiantyn Ralchenko,
Fractional Brownian Motion
Cover
Notations
Introduction
1 Projection of fBm on the Space of Martingales
1.1. fBm and its integral representations
1.2. Formulation of the main problem
1.3. The lower bound for the distance between fBm and Gaussian martingales
1.4. The existence of minimizing function for the principal functional
1.5. An example of the principal functional with infinite set of minimizing functions
1.6. Uniqueness of the minimizing function for functional with the Molchan kernel and H ∈ ( ∈ (, 1)
1.7. Representation of the minimizing function
1.8. Approximation of a discrete-time fBm by martingales
1.9. Exercises
2 Distance Between fBm and Subclasses of Gaussian Martingales
2.1. fBm and Wiener integrals with power functions
2.2. The comparison of distances between fBm and subspaces of Gaussian martingales
2.3. Distance between fBm and class of “similar” functions
2.4. Distance between fBm and Gaussian martingales in the integral norm
2.5. Distance between fBm with Mandelbrot–Van Ness kernel and Gaussian martingales
2.6. fBm with the Molchan kernel and H ∈ (0, ∈ (0, ), in relation to Gaussian martingales
2.7. Distance between the Wiener process and integrals with respect to fBm
2.8. Exercises
3 Approximation of fBm by Various Classes of Stochastic Processes
3.1. Approximation of fBm by uniformly convergent series of Lebesgue integrals
3.2. Approximation of fBm by semimartingales
3.3. Approximation of fBm by absolutely continuous processes
3.4. Approximation of multifractional Brownian motion by absolutely continuous processes
3.5. Exercises
Appendix 1: Auxiliary Results from Mathematical, Functional and Stochastic Analysis
A1.1. Special functions
A1.2. Slope functions: monotone rational function
A1.3. Convex sets and convex functionals
A1.4. The Garsia–Rodemich–Rumsey inequality
A1.5. Theorem on normal correlation
A1.6. Martingales and semimartingales
A1.7. Integration with respect to Wiener process and fractional Brownian motions
A1.8. Existence of integrals of the Molchan kernel and its derivatives
Appendix 2: Evaluation of the Chebyshev Center of a Set of Points in the Euclidean Space
A2.1. Circumcenter of a finite set
A2.2. Constrained least squares
A2.3. Dual problem
A2.4. Algorithm for finding the Chebyshev center
A2.5. Pseudocode of algorithms
Appendix 3: Simulation of fBm
A3.1. The Cholesky decomposition method
A3.2. The Hosking method
A3.3. The circulant method
A3.4. Approximate methods
Solutions
References
Index
End User License Agreement
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