Ph.D Thesis | |

Ph.D Student | Maayan Yohai |
---|---|

Subject | Fractional Brownian Motion |

Department | Department of Mathematics |

Supervisor | ASSOCIATE PROF. Eddy Mayer-Wolf |

Full Thesis text |

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.