Oksana Banna & Yuliya Mishura 
Fractional Brownian Motion 
Approximations and Projections

This monograph studies the relationships between fractional Brownian motion (f Bm) and other processes of more simple form. In particular, this book solves the problem of the projection of f Bm 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 f Bm in the uniform integral norm. Numerical results concerning the approximation problem are given. The upper bounds of distances from f Bm to the different subspaces of Gaussian martingales are evaluated and the numerical calculations are involved. The approximations of f Bm 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 f Bm via the Wiener process are established and some new inequalities for Gamma functions, and even for trigonometric functions, are obtained.

Зміст

1. Projection of f Bm on the Space of Martingales.
2. Distance Between f Bm and Subclasses of Gaussian Martingales.
3. Approximation of f Bm by Various Classes of Stochastic Processes.
Appendix 1. Auxiliary Results from Mathematical, Functional and Stochastic Analysis.
Appendix 2. Evaluation of the Chebyshev Center of a Set of Points in the Euclidean Space.
Appendix 3. Simulation of f Bm.

Про автора

Oksana Banna is Assistant Professor at the Department of Economic Cybernetics at Taras Shevchenko National University of Kyiv (KNU) in Ukraine.

Yuliya Mishura is Full Professor and Head of the Department of Probability, Statistics and Actuarial Mathematics at KNU.

Kostiantyn Ralchenko is Associate Professor at the Department of Probability, Statistics and Actuarial Mathematics at KNU.

Sergiy Shklyar is Senior Researcher at the Department of Probability, Statistics and Actuarial Mathematics at KNU.