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，并为金融业带来了创新的应用。在最近的金融危机之后，允许重尾分布或不做概率假设的新金融模型在学术界和工业界都受到了极大的关注。在这一框架内，某些概率工具已无法使用，需要扩大数学工具包。该项目旨在通过应用巴拿赫格理论的技术来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测度的框架，其中概率定律是根据有序结构来理解的。 拟议的项目将涵盖风险度量和定价理论的公理化理论的主题，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，以及回报（定价），是金融和保险中无处不在的主题。由于风险计量和定价依赖于建模假设，而这些假设的不确定性会带来误差和不准确性，因此，大量研究致力于将这种误判建立在数学上。我预计，提出的Banach格方法将服务于这一角色，预期的发展和结果将在功能分析和数学金融的研究人员的参考价值。

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.