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- W3124355284 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 ($1leq p" @default.
- W3124355284 created "2021-02-01" @default.
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- W3124355284 date "2011-03-01" @default.
- W3124355284 modified "2023-09-27" @default.
- W3124355284 title "Recent progress in random metric theory and its applications to conditional risk measures" @default.
- W3124355284 hasPublicationYear "2011" @default.
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