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Fractional Brownian Motion

Author Oksana Banna , Yuliya Mishura , Kostiantyn Ralchenko
Release Date: 2019/04/01
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Book Description

This monograph studies the relationships between fractional Brownian motion (fBm) and other processes of more simple form. In particular, this book solves the problem of the projection of fBm onto the space of Gaussian martingales that can be represented as Wiener integrals with respect to a Wiener process. It is proved that there exists a unique martingale closest to fBm in the uniform integral norm. Numerical results concerning the approximation problem are given. The upper bounds of distances from fBm to the different subspaces of Gaussian martingales are evaluated and the numerical calculations are involved. The approximations of fBm by a uniformly convergent series of Lebesgue integrals, semimartingales and absolutely continuous processes are presented.

As auxiliary but interesting results, the bounds from below and from above for the coefficient appearing in the representation of fBm via the Wiener process are established and some new inequalities for Gamma functions, and even for trigonometric functions, are obtained.

Table of Contents

  1. Cover
  2. Notations
  3. Introduction
  4. 1 Projection of fBm on the Space of Martingales
    1. 1.1. fBm and its integral representations
    2. 1.2. Formulation of the main problem
    3. 1.3. The lower bound for the distance between fBm and Gaussian martingales
    4. 1.4. The existence of minimizing function for the principal functional
    5. 1.5. An example of the principal functional with infinite set of minimizing functions
    6. 1.6. Uniqueness of the minimizing function for functional with the Molchan kernel and H ∈ ( ∈ (, 1)
    7. 1.7. Representation of the minimizing function
    8. 1.8. Approximation of a discrete-time fBm by martingales
    9. 1.9. Exercises
  5. 2 Distance Between fBm and Subclasses of Gaussian Martingales
    1. 2.1. fBm and Wiener integrals with power functions
    2. 2.2. The comparison of distances between fBm and subspaces of Gaussian martingales
    3. 2.3. Distance between fBm and class of “similar” functions
    4. 2.4. Distance between fBm and Gaussian martingales in the integral norm
    5. 2.5. Distance between fBm with Mandelbrot–Van Ness kernel and Gaussian martingales
    6. 2.6. fBm with the Molchan kernel and H ∈ (0, ∈ (0, ), in relation to Gaussian martingales
    7. 2.7. Distance between the Wiener process and integrals with respect to fBm
    8. 2.8. Exercises
  6. 3 Approximation of fBm by Various Classes of Stochastic Processes
    1. 3.1. Approximation of fBm by uniformly convergent series of Lebesgue integrals
    2. 3.2. Approximation of fBm by semimartingales
    3. 3.3. Approximation of fBm by absolutely continuous processes
    4. 3.4. Approximation of multifractional Brownian motion by absolutely continuous processes
    5. 3.5. Exercises
  7. Appendix 1: Auxiliary Results from Mathematical, Functional and Stochastic Analysis
    1. A1.1. Special functions
    2. A1.2. Slope functions: monotone rational function
    3. A1.3. Convex sets and convex functionals
    4. A1.4. The Garsia–Rodemich–Rumsey inequality
    5. A1.5. Theorem on normal correlation
    6. A1.6. Martingales and semimartingales
    7. A1.7. Integration with respect to Wiener process and fractional Brownian motions
    8. A1.8. Existence of integrals of the Molchan kernel and its derivatives
  8. Appendix 2: Evaluation of the Chebyshev Center of a Set of Points in the Euclidean Space
    1. A2.1. Circumcenter of a finite set
    2. A2.2. Constrained least squares
    3. A2.3. Dual problem
    4. A2.4. Algorithm for finding the Chebyshev center
    5. A2.5. Pseudocode of algorithms
  9. Appendix 3: Simulation of fBm
    1. A3.1. The Cholesky decomposition method
    2. A3.2. The Hosking method
    3. A3.3. The circulant method
    4. A3.4. Approximate methods
  10. Solutions
  11. References
  12. Index
  13. End User License Agreement
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