Banach Spaces: Theory and Application

本提案包含几何泛函分析中的几个公开问题。此外，它还打算将泛函分析的概念应用于金融数学中的解，特别是期权定价理论。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，作者打算使用并推广一些方法，这些方法导致了空间的构造，其中每个算子都是恒等式的倍数的奇异扰动。近年来，Banach空间的渐近性质及其与同构性质的关系越来越受到人们的关注。例如，最近引入的一致渐近凸性和一致渐近光滑性就是这样的性质。这一建议的作者试图找到允许一致渐近凸和光滑重排的同构性质的充要条件。在与R.Gardner和A.Koldobsky合作的一项工作中，作者应用调和分析的概念找到了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。金融学的一个核心问题是找到取决于标的证券的期权的公平价格。这一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。本文旨在研究期权定价的稳健性问题，即期权价格对标的股票模型的连续依赖。研究Banach空间及其几何，因为它们为研究动力系统、微分方程以及最近发现的金融资产定价提供了自然的框架。拟议的项目涉及Banach空间的几何问题，它们之间的算子，以及在金融的数学理解中的应用。

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.