Methods for analysis of functionals on Gaussian self similar processes

PhD Thesis

Bhatia, Hirdyesh. 2018. Methods for analysis of functionals on Gaussian self similar processes . PhD Thesis Doctor of Philosophy. University of Southern Queensland.

Methods for analysis of functionals on Gaussian self similar processes

TypePhD Thesis
AuthorBhatia, Hirdyesh
SupervisorAddie, Ron
Stepanyants, Yury
Institution of OriginUniversity of Southern Queensland
Qualification NameDoctor of Philosophy
Number of Pages131
Digital Object Identifier (DOI)

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.

KeywordsRamer's theorem, finitely additive measures, P-measurable derivative, consistent stochastic flows, transport equation, fractional Brownian motion
ANZSRC Field of Research 2020490510. Stochastic analysis and modelling
490506. Probability theory
Byline AffiliationsSchool of Agricultural, Computational and Environmental Sciences
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