Complete Monotone Quasiconcave Duality

We introduce a notion of complete monotone quasiconcave duality and we show that it holds for important classes of quasiconcave functions.Quasiconcavity, Quasiconvexity, Duality, Indirect Utility

Uncertainty Averse Preferences

We study uncertainty averse preferences, that is, complete and transitive preferences that are convex and monotone. We establish a representation result, which is at same time general and rich in structure. Many objective functions commonly used in applications are special cases of this representation.ambiguity aversion, games against nature, model uncertainty, smooth ambiguity preferences, variational preferences

Dynamic Assessment Indices

This paper provides a unified framework, which allows, in particular, to study the structure of dynamic monetary risk measures and dynamic acceptability indices. The main mathematical tool, which we use here, and which allows us to significantly generalize existing results is the theory of $L^0$-modules. In the first part of the paper we develop the general theory and provide a robust representation of conditional assessment indices, and in the second part we apply this theory to dynamic acceptability indices acting on stochastic processes.Comment: 39 page

Set-valued convex compositions

We study the composition of two set-valued functions defined on locally convex topological linear spaces. We assume that these functions map into certain complete lattices of sets that have been used to establish a conjugation theory for set-valued functions in the literature. Our main result is a formula for the conjugate of the composition in terms of the conjugates of the ingredient functions. As a special case, when the composition is proper and has further regularity, our formula yields a dual representation for the composition. The proof of the main result uses Lagrange duality and minimax theory in a nontrivial way.Comment: 18 page

Risk Measures: Rationality and Diversification

When there is uncertainty about interest rates (typically due to either illiquidity or defaultability of zero coupon bonds) the cash- additivity assumption on risk measures becomes problematic. When this assumption is weakened, to cash-subadditivity for example, the equivalence between convexity and the diversication principle no longer holds. In fact, this principle only implies (and it is implied by) quasiconvexity. For this reason, in this paper quasiconvex risk measures are studied. We provide a dual characterization of quasiconvex cash-subadditive risk measures and we establish necessary and sufficient conditions for their law invariance. As a byproduct, we obtain an alternative characterization of the actuarial mean value premium principle.Risk Measures, Diversification, Cash-subadditivity, Quasiconvexity, Law-invariance, Mean Value Premium Principle

Economies With Many Commodities

We discuss the two fundamental theorems of welfare economics in the context of the Arrow-Debreu-McKenzie model with an infinite dimensional commodity space. As an application, we prove the existence of competitive equilibrium in the standard single agent growth model