随机凸分析(Ⅰ):随机局部凸模中的分离性以及Fenchel-Moreau对偶性  

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

作　　者：郭铁信[1] 赵世恩[2] 曾小林[3] 

机构地区：[1]中南大学数学与统计学院,长沙410083 [2]首都师范大学初等教育学院,北京100048 [3]重庆工商大学数学与统计学院,重庆400067

出　　处：《中国科学：数学》2015年第12期1961-1980,共20页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11171015和11401399);北京市自然科学基金(批准号:1144008)资助项目

摘　　要：为了给出条件风险度量模途径一个牢固的分析基础,本文的目标是在同时考虑(ε,λ)-拓扑和局部L^0-凸拓扑的条件下,在随机局部凸模上建立完整的随机凸分析.本文致力于研究随机局部凸模中的分离性以及Fenchel-Moreau对偶性.主要结果是给出在这两种拓扑下随机局部凸模的两类随机共轭空间的精确关系,这保证了不但可以彻底解决单点集和一个闭L^0-凸集的分离定理,而且可以在随机局部凸模上建立完整的Fenchel-Moreau对偶表示定理.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 L^0-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 L^0-convex subset but also establish the complete Fenchel-Moreau duality theorems in random locally convex modules under the two kinds of topologies.

关 键 词：随机局部凸模 (ε λ)-拓扑 局部L^0-凸拓扑 随机共轭空间 分离定理 下半连续L^0-凸函数 Fenchel-Moreau对偶性 

分 类 号：O189[理学—数学]