M.Sc Student M.Sc Thesis Maayan Yohai Fractional Brownian Motion Department of Mathematics ASSOCIATE PROF. Eddy Mayer-Wolf

Abstract

The main objective of this thesis is to study the covariance function of a process obtained by integrating, with respect to fractional Brownian motion and in the divergence integral sense, certain types of processes. The fractional Brownian motion is a generalization of ordinary Brownian motion, used to model phenomena in a wide range of fields in which self similarity arises (for example, hydrology and stocks).

The first part will concentrate on applying analytic tools to the Malliavin calculus general setting in order to find an integral expression of the aforementioned covariance function. This will require an extensive development the abstract Malliavin derivative to obtain the integration kernel of this derivative (thought of as a Hilbert Schmidt operator) for these processes, as well as an understanding of the relevant Hilbert spaces quantities arising from Malliavin calculus in terms of this kernel.

The second part is motivated by the Tanaka formula, in which the integral of the sign function with respect to fractional Brownian motion appears. For the case of ordinary Brownian motion, this new process turns out to be a Brownian motion as well. The tools developed for finding covariance functions of such processes - actually, for smoother processes, so an approximation is required - will be applied to this process in order to conclude that an H-fractional Brownian motion with H>1/2 does not yield another fractional Brownian motion when the integral of its sign with respect to itself is considered. We will avoid the use of multiple Wiener integrals (the chaos decomposition) throughout the proofs, and will instead try to follow the natural course of the structures of the underlying operators for the covariance calculations.