Integration of the deterministic functions with respect to fractional Brownian motion

Abstract

In this thesis, definition and the characteristic properties of fractional Brownian motion are presented and the general idea for the integration of deterministic functions is discussed with a specific class of integrands. First, some notions and facts from probability theory are introduced. The definition and basic properties of Gaussian random variables and processes are discussed and their relation with the self similar, stationary processes is given. Moreover, covariance function of the self similar Gaussian processes with stationary increments is characterized as in Embrechts and Maejima’s book. Next, we give two representations of fractional Brownian motion. One is defined as a stochastic integral with respect to Brownian motion as in Embrechts and Maejima’s book and the other with the fractional integral as Pipiras and Taqqu do. Then we consider a class of deterministic integrands for the case H > 1/2 which is given by Kleptsyna, LeBreton and Roubaud, and we discuss its completeness. Finally, an example of a complete class of integrands for the case H < 1/2 is introduced as Pipiras and Taqqu do.