数学
有界函数
凸集
订单(交换)
规范(哲学)
组合数学
正多边形
弱拓扑(极性拓扑)
凸函数
次导数
双对
西格玛
离散数学
纯数学
凸优化
拓扑空间
数学分析
一般拓扑结构
功能分析
物理
几何学
扩展拓扑
经济
拓扑张量积
基因
化学
量子力学
生物化学
法学
政治学
财务
作者
Niushan Gao,Denny H. Leung,Foivos Xanthos
出处
期刊:Cornell University - arXiv
日期:2016-01-01
被引量:10
标识
DOI:10.48550/arxiv.1610.08806
摘要
Let $(\Phi,\Psi)$ be a conjugate pair of Orlicz functions. A set in the Orlicz space $L^\Phi$ is said to be order closed if it is closed with respect to dominated convergence of sequences of functions. A well known problem arising from the theory of risk measures in financial mathematics asks whether order closedness of a convex set in $L^\Phi$ characterizes closedness with respect to the topology $\sigma(L^\Phi,L^\Psi)$. (See [26, p.3585].) In this paper, we show that for a norm bounded convex set in $L^\Phi$, order closedness and $\sigma(L^\Phi,L^\Psi)$-closedness are indeed equivalent. In general, however, coincidence of order closedness and $\sigma(L^\Phi,L^\Psi)$-closedness of convex sets in $L^\Phi$ is equivalent to the validity of the Krein-Smulian Theorem for the topology $\sigma(L^\Phi,L^\Psi)$; that is, a convex set is $\sigma(L^\Phi,L^\Psi)$-closed if and only if it is closed with respect to the bounded-$\sigma(L^\Phi,L^\Psi)$ topology. As a result, we show that order closedness and $\sigma(L^\Phi,L^\Psi)$-closedness of convex sets in $L^\Phi$ are equivalent if and only if either $\Phi$ or $\Psi$ satisfies the $\Delta_2$-condition. Using this, we prove the surprising result that: \emph{If (and only if) $\Phi$ and $\Psi$ both fail the $\Delta_2$-condition, then there exists a coherent risk measure on $L^\Phi$ that has the Fatou property but fails the Fenchel-Moreau dual representation with respect to the dual pair $(L^\Phi, L^\Psi)$}. A similar analysis is carried out for the dual pair of Orlicz hearts $(H^\Phi,H^\Psi)$.
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