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启发了一些美丽的数学，并为金融业产生了创新的应用。在最近的金融危机之后，允许厚尾分布或不进行概率假设的新金融模型在学术界和工业界都获得了极大的关注。在这一框架中，某些概率工具变得不可用，出现了对扩展数学工具包的需求。该项目旨在通过应用Banach格子理论中的技术来解决现代数学金融理论的基本问题。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量的框架，其中概率定律是根据有序结构来理解的。*建议的项目将涵盖风险度量公理理论和定价理论的主题，以及一些纯粹的数学问题，这些问题是由金融建模的紧迫挑战引起的。风险和收益(定价)在金融和保险中是一个无处不在的主题。由于风险衡量和定价依赖于建模假设，而建模假设的不确定性会带来错误和不准确，因此大量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Banach格子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.*****