On random convex analysis -- the analytic foundation of the module approach to conditional risk measures

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 $(\epsilon ,\lambda )$--topology and the locally $L^0$-- convex topology). Then, we make use of the advantage of the $(\epsilon ,\lambda )$--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 ($1\le p\le +\infty$) can be uniquely extended to an $L^0$--convex $L_{\mathcal{F}}^p(\mathcal{E})$--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 $L_{\mathcal{F}}^p(\mathcal{E})$--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.

69 pages

References in corpus (1)

• Recent progress in random metric theory and its applications to conditional risk measures

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