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

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 ($1\le p\le +\mathrm{\infty }$) 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.