Applications of Functional Analysis in Foundational Aspects of Mathematical Finance

泛函分析在数学金融基础方面的应用

• 批准号: RGPIN-2019-05518
• 负责人: Xanthos, Foivos
• 金额: $ 1.09万
• 依托单位: Ryerson University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=738344
• 关键词: Applications; Functional; Analysis; Foundational; Aspects

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.

概率理论与数学金融之间的相互作用启发了一些美丽的数学，并为金融行业产生了创新的应用。在最近的金融危机之后，新的财务模式允许进行重型尾部分布或没有做出概率假设，在学术界和工业中都引起了极大的关注。在此框架中，某些概率工具已经变得不可用，并且已经出现了扩展的数学工具包。该项目旨在通过应用Banach Lattices理论的技术来解决现代数学金融理论的基本问题。这是一个越来越多的功能分析领域，它为功能空间提供了无衡量的框架，其中从有序结构方面理解了概率定律。 拟议的项目将涵盖风险措施和定价理论理论的主题，以及一些纯粹的数学问题，这些问题是通过紧迫的财务建模挑战所激发的。风险以及回报（定价）是金融和保险中无处不在的主题。由于风险测量和定价依赖于建模假设，其不确定性引入了错误和不准确，因此广泛的研究一直致力于将这种错误估计以数学上的良好为基础。我预计拟议的Banach晶格方法将发挥这种作用，预期的发展和结果将对研究人员在功能分析和数学金融中具有参考价值。