MAXIMUM LEBESGUE EXTENSION OF CONVEX RISK MEASURES

Abstract

Given a convex risk measure on $L^infty$ having the Lebesgue property, we construct a solid space of random variables on which the original risk measure is extended preserving the Lebesgue property (on the entire space). This space is an order-continuous Banach lattice, and is maximum among all solid spaces admitting such a regular extension. We then characterize the space in terms of uniform integrability of certain families. As a byproduct, we present a generalization of Jouini-Schachermayer-Touzi’s theorem on the weakcompactness characterization of Lebesgue property, which is valid for any solid vector spaces of random variables, and does not require any topological property of the space.