Reflexive Cones

Reflexive cones in Banach spaces are cones with weakly compact intersection with the unit ball. In this paper we study the structure of this class of cones. We investigate the relations between the notion of reflexive cones and the properties of their bases. This allows us to prove a characterization of reflexive cones in term of the absence of a subcone isomorphic to the positive cone of \ell_{1}. Moreover, the properties of some specific classes of reflexive cones are investigated. Namely, we consider the reflexive cones such that the intersection with the unit ball is norm compact, those generated by a Schauder basis and the reflexive cones regarded as ordering cones in a Banach spaces. Finally, it is worth to point out that a characterization of reflexive spaces and also of the Schur spaces by the properties of reflexive cones is given.Comment: 23 page

Computation of vector sublattices and minimal lattice-subspaces of R^k. Applications in finance

In this article we perform a computational study of Polyrakis algorithms presented in [12,13]. These algorithms are used for the determination of the vector sublattice and the minimal lattice-subspace generated by a finite set of positive vectors of R^k. The study demonstrates that our findings can be very useful in the field of Economics, especially in completion by options of security markets and portfolio insurance.Comment: 22 page

THE CHEAPEST HEDGE:A PORTFOLIO DOMINANCE APPROACH

Investors often wish to insure themselves against the payoff of their portfolios falling below a certain value. One way of doing this is by purchasing an appropriate collection of traded securities. However, when the derivatives market is not complete, an investor who seeks portfolio insurance will also be interested in the cheapest hedge that is marketed. Such insurance will not exactly replicate the desired insured-payoff, but it is the cheapest that can be achieved using the market. Analytically, the problem of finding a cheapest insuring portfolio is a linear programming problem. The present paper provides an alternative portfolio dominance approach to solving the minimum-premium insurance portfolio problem. This affords remarkably rich and intuitive insights to determining and describing the minimum-premium insurance portfolios.

The cheapest hedge

Abstract Investors often wish to insure themselves against the payoff of their portfolios falling below a certain value. One way of doing this is by purchasing an appropriate collection of traded securities. However, when the derivatives market is not complete, an investor who seeks portfolio insurance will also be interested in the cheapest hedge that is marketed. Such insurance will not exactly replicate the desired insured-payoff, but it is the cheapest that can be achieved using the market. Analytically, the problem of finding a cheapest insuring portfolio is a linear programming problem. The present paper provides an alternative portfolio dominance approach to solving the minimum-premium insurance portfolio problem. This affords remarkably rich and intuitive insights to determining and describing the minimum-premium insurance portfolios

Yudin Cones and Inductive Limit Topologies

A cone C in a vector space has a Yudin basis {ei}i∈I if every c ∈ C can be written uniquely in the form c = Σi∈Iλiei, where λi ≥ 0 for each i ∈ I and λi = 0 for all but finitely many i. A Yudin cone is a cone with a Yudin basis. Yudin cones arise naturally since the cone generated by an arbitrary family {ei}i∈I of linearly independent vectors C = {∑i∈Iλiei: λi ≥ 0 for each I and λi = 0 for all but finitely many i} is always a Yudin cone having the family { ei}iEJ as a Yudin basis. The Yudin cones possess several remarkable order and topological properties. Here is a list of some of these properties. 1. A Yudin cone C is a lattice cone in the vector subspace it generates M = C - C. 2. A closed generating cone in a two-dimensional vector space is always a Yudin cone. 3. If the cone of a Riesz space is a Yudin cone, then the lattice operations of the space are given pointwise relative to the Yudin basis. 4. If a Riesz space has a Yudin cone, then the inductive limit topology generated by the finite dimensional subspaces is a Hausdorff order continuous locally convex-solid topology. 5. In a Riesz space with a Yudin cone the order intervals lie in finite dimensional Riesz subspaces (and so they are all compact with respect to any Hausdorff linear topology on the space). The notion of a Yudin basis originated in studies on the optimality and efficiency of competitive securities markets in the provision of insurance for investors against risk or price uncertainty.• It is a natural extension to incomplete markets of Arrow's notion of a basis for complete markets, i.e., markets where full insurance against risk can be purchased. The obtained results have immediate applications to competitive securities markets. Especially, they are sufficient for establishing the efficiency of stock markets as a means for insuring against risk or price uncertainty

Bases for Cones and Reflexivity

It is proved that a Banach space E is non-reflexive if and only if E has a closed cone with an unbounded, closed, dentable base. If E is a Banach lattice, the same characterization holds with the extra assumption that the cone is contained in E+. This article is also a survey of the geometry (dentability) of bases for cones. Mathematics Subject Classification (1991): 46A25, 46A40, 46B10, 46B22, 46B42 Keywords: Radon-Nikodym property, dentability/unbounded convex sets, reflexivity and semi-reflexivity, ordered topological linear spaces,vector lattices, duality and reflexivity, Krein-Milman, Banach lattices, Banach, banach space, Banach lattice, lattice Quaestiones Mathematicae 24(2) 2001, 165-17

A Characterization of l1+(Γ)

This article is dedicated to the memory of my friend and collaborator, Yuri A bramovich.Abstract unavailable at this time... Mathematics Subject Classification (2000): 46B3, 46B10, 46B42, 46A55.Key words: Cones, bases for cones, quasi-interior points, dentable sets, Schur property.Quaestiones Mathematicae 27(2004), 99-109