The dual representation problem of risk measures
The dual representation problem of risk measures
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风险测度的双重表示问题
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
F. Xanthos
中科院分区：
文献类型：
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作者：
N. Gao;D. Leung;F. Xanthos
本文的目的是对 Banach 格 $X$ 上风险度量和凸函数的对偶表示问题进行全面研究。特别有趣的是 $X$ 是 Orlicz 空间或 Orlicz 心的情况。我们研究的第一部分致力于$(X,X^\sim_n)$对。在此背景下，我们对 $L_\Phi$ 中凸集 $C$ 的阶次封闭性和 $C$ 相对于拓扑 $\sigma(L_\Phi(\mathbb{P}),L_\Psi(\mathbb{P}))$ 的封闭性之间的关系进行了彻底的分析，最终得到以下令人惊讶的结果： \emph{如果 Orlicz 函数 $\Phi$ 及其共轭 $\Psi$ 都失败了$\Delta_2$-条件，则存在一个连贯的风险测度 $\rho:L_{\Phi}(\mathbb{P})\rightarrow (-\infty,\infty]$ 且 Fatou 性质不承认通过 $L_{\Psi}(\mathbb{P})$} 的双重表示。这个结果回答了风险测度理论中长期存在的开放性问题。在我们研究的第二部分，我们引入了uo-连续对偶$X^{\sim}_{uo}$并探索$(X,X^\sim_{uo})$对的表示问题，这部分扩展了[18]中建立的对$(L_{\Phi}(\mathbb{P}),H_{\Psi}(\mathbb{P}))$的表示结果，并补充了Orlicz心脏的风险度量研究。 $H_{\Phi}(\mathbb{P})$ 在[8]中开发，包含了有关 Banach 格中拓扑和阶次相互作用的新结果和进展，这些结果和进展具有独立意义。
The objective of this paper is to present a comprehensive study of the dual representation problem of risk measures and convex functionals on a Banach lattice $X$. Of particular interest is the case where $X$ is an Orlicz space or an Orlicz heart. The first part of our study is devoted to the pair $(X,X^\sim_n)$. In this setting, we present a thorough analysis of the relationship between order closedness of a convex set $C$ in $L_\Phi$ and the closedness of $C$ with respect to the topology $\sigma(L_\Phi(\mathbb{P}),L_\Psi(\mathbb{P}))$, culminating in the following surprising result: \emph{If an Orlicz function $\Phi$ and its conjugate $\Psi$ both fail the $\Delta_2$-condition, then there exists a coherent risk measure $\rho:L_{\Phi}(\mathbb{P})\rightarrow (-\infty,\infty]$ with the Fatou Property that does not admit a dual representation via $L_{\Psi}(\mathbb{P})$}. This result answers a long standing open problem in the theory of risk measures. In the second part of our study, we introduce the concept of the uo-continuous dual $X^{\sim}_{uo}$ and explore the representation problem for the pair $(X,X^\sim_{uo})$. This part extends the representation result for the pair $(L_{\Phi}(\mathbb{P}),H_{\Psi}(\mathbb{P}))$ established in [18] and complements the study of risk measures on an Orlicz heart $H_{\Phi}(\mathbb{P})$ developed in [8]. This paper contains new results and developments on the interplay between topology and order in Banach lattices that are of independent interest.