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

2015/03/30 by Tiexin Guo, Guo, Tiexin, Shien Zhao +3
Decision Sciences · Mathematics · #46A20 #46A22 #46A55 #46H25 #FOS: Mathematics #Functional Analysis (math.FA) #Fuzzy Systems and Optimization #Multi-Criteria Decision Making #Risk and Portfolio Optimization #math.FA #msc:46A20 #msc:46A22 #msc:46A55 #msc:46H25

paper · pdf · doi:10.48550/arxiv.1503.08695

26 pages; this article draws heavily from arXiv:1210.1848v6. arXiv admin note: text overlap with arXiv:1503.08637

openalex publication_date 2015/03/30 · arxiv created 2015/11/10 · arxiv updated 2015/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

To provide a solid analytic foundation for the module approach to conditional risk measures, our purpose is to establish a complete random convex analysis over random locally convex modules by simultaneously considering the two kinds of topologies (namely the (ε,λ)--topology and the locally L⁰-- convex topology). This paper is focused on the part of separation and Fenchel-Moreau duality in random locally convex modules. The key point of this paper is to give the precise relation between random conjugate spaces of a random locally convex module under the two kinds of topologies, which enables us to not only give a thorough treatment of separation between a point and a closed L⁰-convex subset but also establish the complete Fenchel-Moreau duality theorems in random locally convex modules under the two kinds of topologies.

Citations

• On characterization of locally L⁰-convex topologies induced by a family of L⁰-seminorms
• On random convex analysis -- the analytic foundation of the module approach to conditional risk measures
• Random convex analysis (II): continuity and subdifferentiability theorems in L⁰--pre--barreled random locally convex modules

Related

• On random convex analysis
• On random convex analysis -- the analytic foundation of the module approach to conditional risk measures
• A counterexample shows that not every locally L⁰--convex topology is necessarily induced by a family of L⁰--seminorms
• On the Random Conjugate Spaces of a Random Locally Convex Module
• A study of stability in locally L⁰-convex modules and a conditional version of James' compactness theorem