论文标题
在$ C_B $上的混合拓扑中单调函数的较低半持续性
Lower semicontinuity of monotone functionals in the mixed topology on $C_b$
论文作者
论文摘要
本文的主要结果是，在任意波兰空间上的空间上的单调函数下方的连续性$ c_b $是混合拓扑中的较低的半内态。在这种特殊情况下，混合拓扑与双对$（c_b，{\ rm ca}）$的Mackey拓扑结合，其中$ {\ rm ca} $表示所有可计数附加签名的borel borel borel bore的空间。因此，在凸单单调的混合拓扑中，较低的半持续性图$ c_b \ to \ mathbb r $就偶数添加剂度量而言等同于双表示。这种表示在金融中至关重要，例如在风险措施和超级对冲问题的背景下。基于主要结果，研究了基于$ 2 $ - 替代能力的添加措施的定期属性和Choquet积分的双重表示。在第二步中，本文为凸单酮图的家族的混合拓扑中的等效性提供了表征。结果，对于$ c_b $上的每个凸单映射，在本地凸矢量晶格中取值，混合拓扑中的连续性等同于标准有限集的连续性。
The main result of this paper characterizes the continuity from below of monotone functionals on the space $C_b$ of bounded continuous functions on an arbitrary Polish space as lower semicontinuity in the mixed topology. In this particular situation, the mixed topology coincides with the Mackey topology for the dual pair $(C_b,{\rm ca})$, where ${\rm ca}$ denotes the space of all countably additive signed Borel measures of finite variation. Hence, lower semicontinuity in the mixed topology of convex monotone maps $C_b\to \mathbb R$ is equivalent to a dual representation in terms of countably additive measures. Such representations are of fundamental importance in finance, e.g., in the context of risk measures and super hedging problems. Based on the main result, regularity properties of capacities and dual representations of Choquet integrals in terms of countably additive measures for $2$-alternating capacities are studied. In a second step, the paper provides a characterization of equicontinuity in the mixed topology for families of convex monotone maps. As a consequence, for every convex monotone map on $C_b$ taking values in a locally convex vector lattice, continuity in the mixed topology is equivalent to continuity on norm bounded sets.
