 On random convex analysis -- the analytic foundation of the module approach to conditional risk measuresTiexin Guo, Shien Zhao and Xiaolin Zeng Abstract: To provide a solid analytic foundation for the module approach to conditional risk measures, this paper establishes a complete random convex analysis over random locally convex modules by simultaneously considering the two kinds of topologies (namely the $(\varepsilon,\lambda)$--topology and the locally $L^0$-- convex topology). Then, we make use of the advantage of the $(\varepsilon,\lambda)$--topology and grasp the local property of $L^0$--convex conditional risk measures to prove that every $L^{0}$--convex $L^{p}$--conditional risk measure ($1\leq p\leq+\infty$) can be uniquely extended to an $L^{0}$--convex $L^{p}_{\mathcal{F}}(\mathcal{E})$--conditional risk measure and that the dual representation theorem of the former can also be regarded as a special case of that of the latter, which shows that the study of $L^p$--conditional risk measures can be incorporated into that of $L^{p}_{\mathcal{F}}(\mathcal{E})$--conditional risk measures. In particular, in the process we find that combining the countable concatenation hull of a set and the local property of conditional risk measures is a very useful analytic skill that may considerably simplify and improve the study of $L^{0}$--convex conditional risk measures. Date: 2012-10, Revised 2013-03 Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:1210.1848 Access Statistics for this paper More papers in Papers from arXiv.org |
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