LINEAR AND NON-LINEAR PRICE DECENTRALIZATION

The present paper provides compendious and thorough solutions to the price equilibrium existence problem, the second welfare theorem, and the limit theorem on the core of an economy for exchange economies whose commodity space is an arbitrary ordered Frechet space. The motivation comes from economic applications showing the need to bring within the scope of equilibrium theory commodity spaces that are not vector lattice ordered and whose positive cones have empty interior, a typical situation in models of portfolio trading with incomplete markets. Our assumptions are made on the primitive objects fo the economy. Remarkably, the assumptions that we make on the order structure of the commodity space are indispensable. For w-proper economies, these assumptions are both sufficient and necessary for the existence of equilibrium, the second welfare theorem, and the Edgeworth-Walras equivalence theorem. We take advantage of new developments in the theory of ordered vector spaces, in particular the possibility of embedding the price cone into a lattice cone called the super-order dual of the ordered vector space. Therefore, even though the commodity price duality has no lattice structure important lattice theoretic techniques can be applied outside this duality.

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.

Riesz Estimators.

We consider properties of estimators that can be written as vector lattice (Riesz space) operations. Using techniques widely used in economic theory, we study the approximation properties of these estimators. We also provide two algorithms RIESZVAR(i-ii) for consistent parametric estimation of continuous multivariate piecewise linear functions.

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

Continuous piecewise linear functions

The paper studies the function space of continuous piecewise linear functions in the space of continuous functions on the m-dimensional Euclidean space. It also studies the special case of one dimensional continuous piecewise linear functions. The study is based on the theory of Riesz spaces that has many applications in economics. The work also provides the mathematical background to its sister paper Aliprantis, Harris, and Tourky (2006), in which we estimate multivariate continuous piecewise linear regressions by means of Riesz estimators, that is, by estimators of the the Boolean form where X=(X1, X2, ... , Xm) is some random vector, {Ej}j[set membership]J is a finite family of finite sets