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

TL;DR

This paper advances random convex analysis by introducing $L^0$--pre--barreled modules, establishing their properties, and proving continuity and subdifferentiability theorems for $L^0$--convex functions, with applications to conditional risk measures.

Contribution

It introduces the notion of $L^0$--pre--barreled modules and develops their duality theory, providing key theorems for continuity and subdifferentiability in this framework.

Findings

The model space $L^{p}_{cal}(cal)$ is $L^0$--pre--barreled.

Continuity theorems for proper lower semicontinuous $L^0$--convex functions are established.

Subdifferentiability results are proved for $L^0$--pre--barreled modules.

Abstract

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 $L^{p}_{\mathcal{F}}(\mathcal{E})$ 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.