Banach Spaces: Theory and Application

Banach 空间：理论与应用

• 批准号: 0070456
• 负责人: Thomas Schlumprecht
• 金额: $ 7.8万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070456&HistoricalAwards=false
• 关键词: Banach; Spaces; Theory; Application

ABSTRACTThis proposal contains several open problems in Geometrical Functional Analysis. Furthermore it is intended to applyconcepts of Functional Analysis to solvequestions in Financial Mathematics, in particularin the theory of option pricing. A long standing open question in Operator Theory asks whether ornot there exist Banach spaces on which every operator has an invariant subspace. In order to solve this problem the author intends to use and to extend methods which led to the construction of spaces on which every operator is a singular perturbation of a multiple of the identity. In recent years the notion of asymptotic properties of Banach spaces and their connection to isomorphic properties gained increasing attention. Such properties are for example the recently introduced notion of uniform asymptotic convexity and uniform asymptotic smoothness. The author of this proposal intendsto find sufficient and necessary isomorphic properties admittinguniform asymptotic convex and smooth renormings. In a joint work with R. Gardner and A. Koldobsky the author applied concepts of Harmonic Analysis to find connections between certain extremal properties of convex bodies and higher derivatives of their section functions. The author intends to explore this path further to get more inside on other extremal problems. A central question in Finance is to find fair prices of options contingent to an underlying security. Many results in this area rely on tools developed in Stochastic Calculus as well as Functional Analysis. The author intends to investigate the problem of robustness of option pricing, i.e. the continuous dependence of the optionprice on the underlying stock model.Banach spaces and their geometry are studied since they provide the natural framework for studying dynamical systems, differential equations, and,as discovered recently, the pricing of financial assets. The proposed projects deal with problems on the geometry of Banach spaces, operators between them, and applications to the mathematical understanding of finance.

摘要该提案包含几何泛函分析中的几个开放问题。此外，它旨在应用泛函分析的概念来解决金融数学中的问题，特别是期权定价理论中的问题。算子理论中一个长期悬而未决的问题是，是否存在每个算子都具有不变子空间的 Banach 空间。 为了解决这个问题，作者打算使用并扩展一些方法，这些方法导致构建空间，其中每个算子都是多个恒等式的奇异扰动。 近年来，Banach 空间渐近性质的概念及其与同构性质的联系引起了越来越多的关注。 例如，此类属性是最近引入的均匀渐近凸性和均匀渐近平滑性的概念。该提案的作者打算找到允许一致渐近凸和平滑重整的充分且必要的同构性质。在与 R. Gardner 和 A. Koldobsky 的合作中，作者应用调和分析的概念来寻找凸体的某些极值性质与其截面函数的高阶导数之间的联系。 作者打算进一步探索这条道路，以更深入地了解其他极端问题。 金融领域的一个核心问题是找到与基础证券相关的期权的公平价格。该领域的许多结果依赖于随机微积分和泛函分析中开发的工具。作者打算研究期权定价的鲁棒性问题，即期权价格对基础股票模型的连续依赖性。对巴纳赫空间及其几何进行研究，因为它们为研究动力系统、微分方程和，作为最近发现，金融资产的定价。拟议的项目处理巴纳赫空间的几何问题、它们之间的算子以及在金融数学理解中的应用。