Applications of Functional Analysis in Foundational Aspects of Mathematical Finance

概率论和数学金融之间的相互作用激发了一些美丽的数学，并为金融行业带来了创新的应用。在最近的金融危机之后，新的金融模型允许重尾分布，或者不做概率假设，在学术界和工业界都引起了极大的关注。在这个框架中，某些概率工具变得不可用，并且需要一个扩展的数学工具包。本项目旨在运用巴拿赫格理论的技术解决现代数学金融理论的基础问题。这是泛函分析的一个不断发展的领域，它为函数空间提供了一个无度量的框架，其中概率律是根据有序结构来理解的。该项目将涵盖风险度量和定价理论的公理化理论，以及一些由金融建模的紧迫挑战所激发的纯数学问题。风险和收益（定价）一样，是金融和保险领域一个无处不在的主题。由于风险测量和定价依赖于建模假设，而建模假设的不确定性会引入错误和不准确性，因此大量的研究致力于将这种错误计算建立在数学上合理的基础上。我预计所提出的巴拿赫格方法将发挥这一作用，预期的发展和结果将对泛函分析和数学金融的研究人员具有参考价值。

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.