Available at: https://digitalcommons.calpoly.edu/theses/3055
Date of Award
6-2025
Degree Name
MS in Mathematics
Department/Program
Mathematics
College
College of Science and Mathematics
Advisor
Ryan Tully-Doyle
Advisor Department
Mathematics
Advisor College
College of Science and Mathematics
Abstract
The method of kernel embedding of distributions on a set $\Omega$ into a reproducing kernel Hilbert space is a method of studying the space of measures on a set $\Omega$ using Hilbert space geometry. Because positively weighted portfolios can be interpreted as nonnegative probability measures on the space of assets, we are able to apply this technique to portfolio theory. In this thesis, we discuss the theory of "topiarism", the study of positively weighted probability measures on compact sets under the kernel embedding of distributions. Given a specific payoff function $\psi$ on the set of assets $\Omega$, we optimize a specific function $\eta$, called the aesthetic objective, by finding maximizing measures. We review different properties, develop an algorithm for locating the maximizer of the function $\eta$, and discuss applications of this theory to both modern portfolio theory and as a method for rudimentary boundary-finding.