Ioannis A. Polyrakis

Portfolio dominance and optimality in infinite security markets

The most natural way of ordering portfolios is by comparing their payoffs. A portfolio with payoff higher than the payoff of another portfolio is greater in the sense of portfolio dominance than that other portfolio. Portfolio dominance is a lattice order if the supremum and the infimum of any two portfolios are well-defined. We study security markets with infinitely many securities and arbitrary finite portfolio holdings. If portfolio dominance order is a lattice order and has a Yudin basis, then optimal portfolio allocations and equilibria in security markets do exist.

Minimal lattice-subspaces

In this paper the existence of minimal lattice-subspaces of a vector lattice E containing a subset B of E+ is studied (a lattice-subspace of E is a subspace of E which is a vector lattice in the induced ordering). It is proved that if there exists a Lebesgue linear topology τ on E and E+ is τ -closed (especially if E is a Banach lattice with order continuous norm), then minimal lattice-subspaces with τ -closed positive cone exist (Theorem 2.5). In the sequel it is supposed that B = {x1, x2, . . . , xn} is a finite subset of C+(Ω), where Ω is a compact, Hausdorff topological space, the functions xi are linearly independent and the existence of finite-dimensional minimal lattice-subspaces is studied. To this end we define the function β(t) = r(t) ‖r(t)‖1 where r(t) = ( x1(t), x2(t), . . . , xn(t) ) . If R(β) is the range of β and K the convex hull of the closure of R(β), it is proved: (i) There exists an m-dimensional minimal lattice-subspace containing B if and only if K is a polytope of Rn with m vertices (Theorem 3.20). (ii) The sublattice generated by B is an m-dimensional subspace if and only if the set R(β) contains exactly m points (Theorem 3.7). This study defines an algorithm which determines whether a finite-dimensional minimal lattice-subspace (sublattice) exists and also determines these subspaces.

Linear Optimization in C (Ω) and Portfolio Insurance

Suppose that X is a subspace of C ( z ) generated by n linearly independent positive elements of C ( z ). In this article we study the problem of minimization of a positive linear functional p of X in X , under a finite number of linear inequalities. This problem does not have always a solution and if a solution exists we cannot determine it. In this article we show that if X is contained in a finite dimensional minimal lattice-subspace Y of C ( z ) (or equivalently, if X is contained in a finite dimensional minimal subspace Y of C ( z ) with a positive basis) and m = dim Y , then the minimization problem has a solution and we determine the solutions by solving an equivalent linear programming problem in . Finally note that this minimization problem has an important application in the portfolio insurance which was the motivation for the preparation of this article.

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.

Schauder bases in locally solid lattice Banach spaces

Schauder bases in Banach spaces are studied in [5]. In ordered Banach spaces a special type of Schauder bases, the O.P. Schauder bases, are studied because then the properties of ordered spaces can be used.

Cones locally isomorphic to the positive cone of l1(Γ)

Abstract We give necessary and sufficient conditions in order for an infinite-dimensional, closed cone P of a Banach space X to be locally isomorphic to the positive cone l+1(Γ) of l1(Γ).

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.

Spaces with Dual Order-Isomorphic to ℓ1

We establish that an ordered Banach space is order-isomorphic to c0 if and only if it is a σ-Dedekind complete vector lattice and its norm dual is order-isomorphic to ℓ1.

Grothendieck ordered Banach spaces with an interpolation property

In this paper we prove that if E is an ordered Banach space with the countable interpolation property, E has an order unit and E+ is closed and normal, then E is a Grothendieck space; i.e. any weak-star convergent sequence of E∗ is weakly convergent. By the countable interpolation property we mean that for any A,B ⊆ E countable, with A ≤ B, we have A ≤ {x} ≤ B for some x ∈ E.

Nonreplication of Options

In this paper, we study the replication of options in security markets X with a finite number of states. Specifically, we prove that in security markets without binary vectors, for any portfolio, at most options can be replicated where m is the number of states. This is an essential improvement of the result of Baptista where it is proved that the set of replicated options is of measure zero. Additionally, we extend the results of Aliprantis and Tourky on the nonreplication of options by generalizing their condition that markets are strongly resolving. Our results are based on the theory of lattice‐subspaces and positive bases.