A Panorama of Applied Mathematical Problems in Economics

Hung T. Nguyen, Nguyen Ngoc Thach

Abstract

We present a variety of issues in modern economics where applied mathematics is needed. These include economic equilibrium prob- lem where Öxed point theory for Markov operators, and for ordered sets is needed; the emergent Öeld of behavioral economics (Nobel Memorial Prize in Economics 2017) where more mathematical research is needed, for ex- ample, to capture cognitive decision-making, a§ecting economic data, and hence econometric modeling processes. We will focus on a promising venue, namely, quantum probability as a measure of uncertainty, since, unlike Kol- mogorov probability, quantum probability, borrowed from quantum physics, is non-additive and non-commutative, thus avoiding Ellsbergís paradox (of von Neumannís expected utility) and capturing order e§ects of human judge- ments, respectively. Evoking the development of the framework of quantum probability as behavioral probability, as well as a quantum stochastic calculus for, say, Önancial econometrics, we will discuss various important mathemati