Interrupted Brownian motion and other related processes

Tee, Jing Han (2022) Interrupted Brownian motion and other related processes. PhD thesis, London School of Economics and Political Science.

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Abstract

In this thesis, we study the distributional properties of functionals of the Brownian motion. The thesis starts with an analysis of the occupation time process of Brownian motion in which the joint Laplace transforms of the occupation time processes in different regions and their driving Brownian motion are computed for different starting points using martingale methodology. The corresponding joint density functions are also derived. A version of the Brownian motion, called the interrupted Brownian motion is introduced in the next chapter where the paths of the Brownian motion within a certain band are eliminated. Some distributional properties of the interrupted Brownian motion are derived using the perturbation method. The study of the local time at a certain level of the Brownian motion is then investigated using the Feynman-Kac formulas to derive the joint Laplace transforms of the local time evaluated at the first inverse time local time of the Brownian motion. We repeat the procedure for a compound Poisson process with drift. This thesis is concluded with a discussion on hitting and exit times of other diffusion using symmetry methods. In particular, we look at a diffusion related to Nicholson’s integral and another diffusion by conditioning on the Brownian motion.

Item Type:	Thesis (PhD)
Additional Information:	© 2022 Jing Han Tee
Library of Congress subject classification:	Q Science > QA Mathematics
Sets:	Departments > Statistics
Supervisor:	Dassios, Angelos
URI:	http://etheses.lse.ac.uk/id/eprint/4522

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