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

Abstract

The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures. This paper includes eight sections. Section 1 is a longer introduction, which gives a brief introduction to random metric theory, risk measures and conditional risk measures. Section 2 gives the central framework in random metric theory, topological structures, important examples, the notions of a random conjugate space and the Hahn-Banach theorems for random linear functionals. Section 3 gives several important representation theorems for random conjugate spaces. Section 4 gives characterizations for a complete random normed module to be random reflexive. Section 5 gives hyperplane separation theorems currently available in random locally convex modules. Section 6 gives the theory of random duality with respect to the locally $L^0-$convex topology and in particular a characterization for a locally $L^0-$convex module to be $L^0-$pre$-$barreled. Section 7 gives some basic results on $L^0-$convex analysis together with some applications to conditional risk measures. Finally, Section 8 is devoted to extensions of conditional convex risk measures, which shows that every representable $L^{\infty }-$type of conditional convex risk measure and every continuous $L^p-$type of convex conditional risk measure ($1\le p<+\infty$) can be extended to an $L_{\mathcal{F}}^{\infty }(\mathcal{E})-$type of ${\sigma }_{\epsilon ,\lambda }(L_{\mathcal{F}}^{\infty }(\mathcal{E}),L_{\mathcal{F}}^1(\mathcal{E}))-$lower semicontinuous conditional convex risk measure and an $L_{\mathcal{F}}^p(\mathcal{E})-$type of ${\mathcal{T}}_{\epsilon ,\lambda }-$continuous conditional convex risk measure ($1\le p<+\infty$), respectively.

37 pages

References in corpus (1)

• A comprehensive connection between the basic results and properties derived from two kinds of topologies for a random locally convex module

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• A geometric form of the Hahn-Banach extension theorem for L0 linear functions and the Goldstine-Weston theorem in random normed modules
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• The Goldstine-Weston theorem in random normed modules
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• On the Random Conjugate Spaces of a Random Locally Convex Module
• Ekeland's Variational Principle for An ${\bar{L}}^0-$Valued Function on A Complete Random Metric Space