Random convex analysis (II): continuity and subdifferentiability theorems in $L^0$--pre--barreled random locally convex modules

In this paper, we continue to study random convex analysis. First, we introduce the notion of an $L^0$–pre–barreled module. Then, we develop the theory of random duality under the framework of a random locally convex module endowed with the locally $L^0$–convex topology in order to establish a characterization for a random locally convex module to be $L^0$–pre–barreled, in particular we prove that the model space $L_{\mathcal{F}}^p(\mathcal{E})$ employed in the module approach to conditional risk measures is $L^0$–pre–barreled, which forms the most difficult part of this paper. Finally, we prove the continuity and subdifferentiability theorems for a proper lower semicontinuous $L^0$–convex function on an $L^0$–pre–barreled random locally convex module. So the principal results of this paper may be well suited to the study of continuity and subdifferentiability for $L^0$–convex conditional risk measures.