Dual Representation of Quasi-convex Conditional Maps

Abstract

We provide a dual representation of quasi-convex maps $\pi :L_{\mathcal{F}% }\rightarrow L_{\mathcal{G}}$, between two locally convex lattices of random variables, in terms of conditional expectations. This generalizes the dual representation of quasi-convex real valued functions $\pi :L_{\mathcal{F}% }\rightarrow \mathbb{R}$ and the dual representation of conditional convex maps $\pi :L_{\mathcal{F}}\rightarrow L_{\mathcal{G}}.$ These results were inspired by the theory of dynamic measurements of risk and are applied in this context.