Methods for analysis of functionals on Gaussian self similar processes

Methods for analysis of functionals on Gaussian self similar processes

Many engineering and scientific applications necessitate the estimation of statistics of various functionals on stochastic processes. In Chapter 2, Norros et al’s Girsanov theorem for fBm is reviewed and extended to allow for non-unit volatility. We then prove that using method of images to solve the Fokker-Plank/Kolmogorov equation with a Dirac delta initial condition and a Dirichlet boundary condition to evaluate the first passage density, does not work in the case of fBm.

Chapter 3 provides generalisation of both the theorem of Ramer which finds a formula for the Radon-Nikodym derivative of a transformed Gaussian measure and of the Girsanov theorem. A P-measurable derivative of a P-measurable function is defined and then shown to coincide with the stochastic derivative, under certain assumptions, which in turn coincides with the Malliavin derivative when both are defined. In Chapter 4 consistent quasi-invariant stochastic flows are defined. When such a flow transforms a certain functional consistently a simple formula exists for the density of that functional. This is then used to derive the last exit distribution of Brownian motion.

In Chapter 5 a link between the probability density function of an approximation of the supremum of fBm with drift and the Generalised Gamma distribution is established. Finally the self-similarity induced on the distributions of the sup and the first passage functionals on fBm with linear drift are shown to imply the existence of transport equations on the family of these densities as the drift varies.