Banach Spaces: Theory and Application

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

ABSTRACT This proposal contains several open problems in Geometrical Functional Analysis. Furthermore it is intended to applyconcepts of Functional Analysis to solve questions in Financial Mathematics, in particularin the theory of option pricing. A long standing open question in Operator Theory asks whether or not 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 intends to find sufficient and necessary isomorphic properties admitting uniform 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 option price 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.