Banna O. Approximation of fractional Brownian motion by Gaussian martingales involving integrands represented by several functional classes.

The thesis is devoted to the approximation of fractional Brownian motion with the help of Gaussian martingales that can be represented as the stochastic integrals with respect to a Wiener process. It is proved that the distance between fractional Brownian motion and the whole space of such integrals in non-zero. This distance is estimated from below. The values of the distances between fractional Brownian motion and several subspaces of the space of Gaussian martingales are established. In some cases these distances are estimated from above and the numerical calculations are involved. As an auxiliary but interesting results, the bounds from below and from above for the coefficient appearing in the representation of fBm via Wiener process are established and some new inequalities for Gamma-functions are obtained.