Fractional Brownian Motion (Approximations of FBm) : Weak and Strong Approximations and Projections

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. Content: Cover Half-Title Page Title Page Copyright Page Contents 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 2, 1 1.7. Representation of the minimizing function 1.7.1. Auxiliary results1.7.2. Main properties of the minimizing function 1.8. Approximation of a discrete-time fBm by martingales 1.8.1. Description of the discrete-time model 1.8.2. Iterative minimization of the squared distance using alternating 1.8.3. Implementation of the alternating minimization algorithm 1.8.4. Computation of the minimizing function 1.9. Exercises 2. Distance Between fBm and Subclasses of Gaussian Martingales 2.1. fBm and Wiener integrals with power functions 2.1.1. fBm and Wiener integrals with constant integrands 2.1.2. fBm and Wiener integrals involving power integrands with a positive exponent2.1.3. fBm and integrands a(s) with a(s)s-a non-decreasing 2.1.4. fBm and Gaussian martingales involving power integrands with a negative exponent 2.1.5. fBm and the integrands a(s) = a0sa + a1sa+1 2.1.6. fBm and Wiener integrals involving integrands k1 + k2sa 2.2. The comparison of distances between fBm and subspaces of Gaussian martingales 2.2.1. Summary of the results concerning the values of the distances 2.2.2. The comparison of distances 2.2.3. The comparison of upper and lower bounds for the constant cH2.3. Distance between fBm and class of "similar" functions 2.3.1. Lower bounds for the distance 2.3.2. Evaluation 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 2.5.1. Constant function as an integrand 2.5.2. Power function as an integrand 2.5.3. Comparison of Molchan and Mandelbrot-Van Ness kernels 2.6. fBm with the Molchan kernel and H . (0, 1 2), in relation to Gaussian martingales 2.7. Distance between the Wiener process and integrals with respect to fBm2.7.1. Wiener integration with respect to fBm 2.7.2. Wiener process and integrals of power functions 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 3.2.1. Construction and convergence of approximations 3.2.2. Approximation of an integral with respect to fBm by integrals with respect to semimartingales