Magill-type theorems for mappings

Magill's and Rayburn's theorems on the homeomorphism of Stone-Čech remainders and some of their generalizations to the remainders of arbitrary Hausdorff compactifications of Tychonoff spaces are extended to some class of mappings

Compactification-like extensions

Let $X$ be a space. A space $Y$ is called an extension of $X$ if $Y$ contains $X$ as a dense subspace. For an extension $Y$ of $X$ the subspace $Y\backslash X$ of $Y$ is called the remainder of $Y$. Two extensions of $X$ are said to be equivalent if there is a homeomorphism between them which fixes $X$ pointwise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Y\leq Y'$ if there is a continuous mapping of $Y'$ into $Y$ which fixes $X$ pointwise. Let $P$ be a topological property. An extension $Y$ of $X$ is called a $P$-extension of $X$ if it has $P$. If $P$ is compactness then $P$-extensions are called ompactifications. The aim of this article is to introduce and study classes of extensions (which we call compactification-like $P$-extensions, where $P$ is a topological property subject to some mild requirements) which resemble the classes of compactifications of locally compact spaces. We formally define compactification-like $P$-extensions and derive some of their basic properties, and in the case when the remainders are countable, we characterize spaces having such extensions. We will then consider the classes of compactification-like $P$-extensions as partially ordered sets. This consideration leads to some interesting results which characterize compactification-like $P$-extensions of a space among all its Tychonoff $P$-extensions with compact remainder. Furthermore, we study the relations between the order-structure of classes of compactification-like $P$-extensions of a Tychonoff space $X$ and the topology of a certain subspace of its outgrowth $\beta X\backslash X$. We conclude with some applications, including an answer to an old question of S. Mr\'{o}wka and J.H. Tsai: For what pairs of topological properties $P$ and $Q$ is it true that every locally-$P$ space with $Q$ has a one-point extension with both $P$ and $Q$?Comment: 86 page

Existence of pseudo-equilibria in a financial economy

This paper proves the existence of a pseudo-equilibrium in a financial economy with incomplete markets in which the agents may have nonordered preferences. We will use a fixed-point-like theorem of [4] that generalizes the results by Hirsch, Magill, Mas-Colell [18] and Husseini, Lasry, Magill [19] to encompass the framework considered by Gale and Mas-Colell ([14],[15]).Pseudo-equilibrium, incomplete markets, nonordered preferences, fixed-point-like theorems, Grassmann manifold.

Existence of pseudo-equilibria in a financial economy

This paper proves the existence of a pseudo-equilibrium in a financial economy with incomplete markets in which the agents may have nonordered preferences. We will use a fixed-point-like theorem of Bich and Cornet that generalizes the results by Hirsch, Magill, Mas-Colell [18] and Husseini, Lasry, Magill [19] to encompass the framework considered by Gale and Mas-Colell ([14], [15]).Pseudo-equilibrium ; incomplete markets ; nonordered preferences ; fixed-point-like theorems ; Grassmann manifold

General equilibrium and fixed point theory : a partial survey

Focusing mainly on equilibrium existence results, this paper emphasizes the role of fixed point theorems in the development of general equilibrium theory, as well for its standard definition as for some of its extensions.Fixed point, equilibrium, quasiequilibrium, abstract economy, Clarke's normal cone, financial equilibrium, Grassmanian manifold, degree theory.

Arbitrage and Equilibrium with Portfolio Constraints

We consider a multiperiod financial exchange economy with nominal assets and restricted participation, where each agent’s portfolio choice is restricted to a closed, convex set containing zero, as in Siconolfi (1989). Using an approach that dates back to Cass (1984, 2006) in the unconstrained case, we seek to isolate arbitrage-free asset prices that are also quasi-equilibrium or equilibrium asset prices. In the presence of such portfolio restrictions, we need to confine our attention to aggregate arbitrage-free asset prices, i.e., for which there is no arbitrage in the space of marketed portfolios. Our main result states that such asset prices are quasi-equilibrium prices under standard assumptions and then deduce that they are equilibrium prices under a suitable condition on the accessibility of payoffs by agents, i.e., every payoff that is attainable in the aggregate can be marketed through some agent’s portfolio set. This latter result extends previous work by Martins-da-Rocha and Triki (2005).Stochastic Financial exchange economies; Incomplete markets; Financial equilibrium; Constrained portfolios; Multiperiod models; Arbitrage-free asset prices

Arbitrage and Equilibrium with Portfolio Constraints

We consider a multiperiod financial exchange economy with nominal assets and restricted participation, where each agent's portfolio choice is restricted to a closed, convex set containing zero, as in Siconolfi (1989). Using an approach that dates back to Cass (1984, 2006) in the unconstrained case, we seek to isolate arbitrage-free asset prices that are aloso quasi-equilibrium or equilibrium asset prices. In the presence of such portfolio restrictions, we need to confine our attention to aggregate arbitrage-free asset prices, i.e., for which there is no arbitrage in the space of marketed portfolios. Our main result states that such asset prices are quasi-equilibrium prices under standard assumptions and then deduce that they are equilibrium prices under a suitable condition on the accessibility of payoffs by agents, i.e., every payoff that is attainable in the aggregate can be marketed through some agent's portfolio set. This latter result extends previous work by Martins-da-Rocha and Triki (2005).Stochastic financial exchange economies, incomplete markets, financial equilibrium, constrained portfolios, multiperiod models, arbitage-free asset prices.