Ph.D Thesis
Ph.D Student Yohai Maayan Fractional Brownian Motion Department of Mathematics Professor Mayer-Wolf Eddy

Abstract

This thesis is concerned with a type of stochastic process called fractional Brownian motion, which can be used to model natural as well as non-natural phenomena. It is not one but rather a family of processes, since it depends on a self-similarity index H called the Hurst parameter. It generalises the Brownian motion, which is the case H=1/2, thus allowing for non-zero correlation between disjoint increments. As a result, it is more suitable than Brownian motion for modeling certain systems such as fluctuations in solids, stock markets, and more.

In the first part of the thesis, we develop an analytic formula for the covariance between certain types of stochastic integrals called divergence integrals with respect to a multidimensional fractional Brownian motion with Hurst parameter H>1/2. Unlike the case of Brownian motion, where the Itô stochastic integral for adapted processes has an isometry property, the divergence integral possesses no such property. However, there is an abstract formula for the covariance in the setting of the stochastic calculus of variations (the Malliavin calculus). We identify the operator quantities for a large range of integrands which are point functions of the fractional Brownian motion.

We show an interesting application of the formula for processes which are related to fractional Bessel processes.

In the second part of the Thesis, we compute the Onsager-Machlup functional for the solution of a stochastic differential equation with respect to a fractional Brownian motion with Hurst parameter H<1/2. This functional compares the probability that the solution lies in a small `ball' in the space of paths with respect to the supremum or Hölder norms, centered at a deterministic path, with the probability that the fractional Brownian motion itself lies in such a small ball.

To facilitate the analysis described above, we also study two underlying Hilbert path spaces associated with the fractional Brownian motion: its Cameron-Martin space and a related space of paths which play the role of deterministic integrands. We prove several results pertaining to each one of them and clarify the connection between them. While useful as tools for our main results, we believe these results are of independent interest.