Choquet integral and conditional expectation in uncertainty space

PhD Dissertation Defense
Choquet integral and conditional expectation in uncertainty space

by Michael B. Frondoza
PhD Mathematics Candidate

Date: Friday, 20 May 2022
Time: 4:30 pm
Venue: Online

Advisers:
Dr. Elvira P. de Lara-Tuprio & Dr. Richard B. Eden
Ateneo de Manila University

Panelists:
Dr. Jayrold Arcede
Caraga Stage University
Dr. Emmanuel A. Cabral
Ateneo de Manila University
Dr. Kristine Joy Carpio
De La Salle University
Dr. Jose Maria L. Escaner IV
University of the Philippines Diliman

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 in- equalities.

We next define conditional expectations with respect to σ-algebras, similar to the standard definition of con- ditional 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 σ-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 expec- tations and prove some of their properties.

Key Words: uncertain measure, Choquet integral, conditional expectation, Radon-Nikodym Theorem, atoms of a σ-algebra