Choquet Integral and Conditional Expectation in Uncertainty Space
Date of Award
7-1-2022
Document Type
Dissertation
Degree Name
Doctor of Philosophy in Mathematics
First Advisor
Elvira P. de Lara-Tuprio, PhDRichard B. Eden, PhD
Abstract
Uncertain measures and uncertain variables were defined by Baoding Liu in 2007. This dissertation defines a Choquet integral with respect to an uncertain measure. Choquet integrable uncertain variables are then introduced, along with properties of their integrals such as a sub-additivity theorem and some standard inequalities.
We next define conditional expectations with respect to -algebras, similar to the standard definition of conditional expectations in probability spaces. In our current setting, a version of the Radon-Nikodym Theorem for uncertain measures is used to show the existence of conditional expectations of non-negative uncertain variables. The definition is then extended to uncertain variables of arbitrary sign. Properties of conditional expectations based on this definition are presented.
Finally, we provide another way of defining a conditional expectation of an uncertain variable X with respect to a algebra G, as a G-measurable function Y which minimizes the integral of (X -Y )2. The development assumes a finite sample space and finitely many atoms for G. We justify the existence of conditional expectations and prove some of their properties.
Keywords: uncertain measure, Choquet integral, conditional expectation, Radon-Nikodym Theorem, atoms of a -algebra
Recommended Citation
Frondoza, Michael B., (2022). Choquet Integral and Conditional Expectation in Uncertainty Space. Archīum.ATENEO.
https://archium.ateneo.edu/theses-dissertations/982
