Abstracts

Convex risk measures on $L^\infty$ have been studied in various aspects. Among many other fine properties, it is known that a risk measure on $L^\infty$ has the so-called robust representation by probability measures if and only if it has the Fatou property (order lower semicontinuity), and for such a risk measure, there is equivalence between (1) the so-called Lebesgue property (continuity w.r.t. the dominated a.s. convergence),(2) the weak compactness of all the sublevel sets of the conjugate, and (3) the attainment of the supremum in the robust representation (Jouini-Schachermayer-Touzi's theorem). Each of equivalent properties has importance in application, and the implication (3) $\Rightarrow$ (2) may be viewed as a partial generalization of perturbed James' theorem. Recently, Orihuela and Ruiz Galán obtained a similar equivalence for risk measures on certain class of Orlicz spaces. In this talk, we provide this type of equivalence for monotone convex functions on solid subspaces of $L^0$, which improve the one by Orihuela and Ruiz Galán, with a much simpler proof, and unifies several other related results. We then discuss applications and implications in financial mathematics.