Date of Award

May 2015

Degree Type

Thesis

Degree Name

Master of Science

Department

Mathematics

First Advisor

Richard H. Stockbridge

Committee Members

Eric S. Key, Chao Zhu

Keywords

American Option, Geometric Brownian Motion, Integral Representation for Excessive Function, Optimal Investment Problem, Optimal Stopping Problem, Riesz Representation

Abstract

In this thesis we summarize results about optimal stopping problems analyzed with

the Riesz representation theorem. Furthermore we consider two examples: Firstly

the optimal investment problem with an underlying d-dimensional geometric Brow-

nian motion. We derive formulas for the optimal stopping boundaries for the one-

and two-dimensional cases and we find a numerical approximation for the boundary

in the two-dimensional problem. After this we change the focus to a space-time

one-dimensional geometric Brownian motion with finite time horizon. We use the

Riesz representation theorem to approximate the optimal stopping boundaries for

three financial options: the American Put option, American Cash-or-Nothing option

and the American Asset-or-Nothing option.

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