Fractional Brownian Motion: Approximations and Projections [Hardcover]

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.

Notations ix

Introduction xiii

Chapter 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 5

1.3. The lower bound for the distance between fBm and Gaussian martingales 8

1.4. The existence of minimizing function for the principal functional 10

1.5. An example of the principal functional with infinite set of minimizing functions 12

1.6. Uniqueness of the minimizing function for functional with the Molchan kernel and H  (1/2,1) 17

1.7. Representation of the minimizing function 21

1.7.1. Auxiliary results 21

1.7.2. Main properties of the minimizing function 28