Applications of Functional Analysis in Foundational Aspects of Mathematical Finance

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The interplay between Probability Theory and Mathematical Finance has inspired some beautiful mathematics and produced innovative applications for the financial industry. In the aftermath of the recent financial crises, new financial models that allow for heavy tailed distributions, or make no probabilistic assumptions, have gained tremendous attention both in academia and industry. In this framework, certain probabilistic tools have become unavailable, and a need for an expanded mathematical toolkit has emerged. This project aims to address foundational problems of modern Mathematical Finance theory by applying techniques from the theory of Banach lattices. This is a growing area of Functional Analysis that provides a measure-free framework for function spaces, where probabilistic laws are understood in terms of ordered structures. The proposed project will cover topics in the axiomatic theory of risk measures and pricing theory as well as some pure mathematical problems that are motivated by pressing challenges of financial modeling. Risk, as well as return (pricing), is a ubiquitous subject in finance and insurance. As risk measurement and pricing relies on modeling assumptions, whose uncertainty introduces errors and inaccuracies, a broad stream of research has been devoted to putting this miscalculation on a mathematically sound basis. I anticipate that the proposed Banach lattice approach will serve this role, and the expected developments and results will be of referential value to researchers in both Functional Analysis and Mathematical Finance.