Random convex analysis (II): continuity and subdifferentiability theorems in L⁰--pre--barreled random locally convex modules

2015/03/31 by Tiexin Guo, Shien Zhao, Xiaolin Zeng
Mathematics · #math.FA #msc:46A08 #msc:46A20 #msc:46H25 #msc:46T20

paper · pdf · doi:10.1360/n012015-00042

published as Sci Sin Math, 2015, 45: 647-662 · 19 pages. this article draws heavily from arXiv:1210.1848v6

arxiv created 2015/11/10 · arxiv updated 2015/11/11

Abstract

In this paper, we continue to study random convex analysis. First, we introduce the notion of an L⁰--pre--barreled module. Then, we develop the theory of random duality under the framework of a random locally convex module endowed with the locally L⁰--convex topology in order to establish a characterization for a random locally convex module to be L⁰--pre--barreled, in particular we prove that the model space L^p_F(E) employed in the module approach to conditional risk measures is L⁰--pre--barreled, which forms the most difficult part of this paper. Finally, we prove the continuity and subdifferentiability theorems for a proper lower semicontinuous L⁰--convex function on an L⁰--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 L⁰--convex conditional risk measures.

Citations

• On random convex analysis -- the analytic foundation of the module approach to conditional risk measures

Cited by

• Random convex analysis (I): separation and Fenchel-Moreau duality in random locally convex modules