Recent progress in random metric theory and its applications to conditional risk measures
 TieXin Guo
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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 HahnBanach 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}prebarreled. 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 ^{∞}type of conditional convex risk measure and every continuous L ^{ p }type of convex conditional risk measure (1 ≤ p < +∞) can be extended to an \( L_\mathcal{F}^\infty \left( \mathcal{E} \right) \) type of σ _{ ε,λ } \( \sigma _{\varepsilon ,\lambda } \left( {L_\mathcal{F}^\infty \left( \mathcal{E} \right),L_\mathcal{F}^1 \left( \mathcal{E} \right)} \right) \) lower semicontinuous conditional convex risk measure and an \( L_\mathcal{F}^\infty \left( \mathcal{E} \right) \) type of \( \mathcal{T}_{\varepsilon ,\lambda } \) continuous conditional convex risk measure (1 ≤ p < +∞), respectively.
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 Title
 Recent progress in random metric theory and its applications to conditional risk measures
 Journal

Science China Mathematics
Volume 54, Issue 4 , pp 633660
 Cover Date
 20110401
 DOI
 10.1007/s1142501141896
 Print ISSN
 16747283
 Online ISSN
 18691862
 Publisher
 SP Science China Press
 Additional Links
 Topics
 Keywords

 random normed module
 random inner product module
 random locally convex module
 random conjugate space
 L 0convex analysis
 conditional risk measures
 46A22
 46A25
 46H25
 47H40
 52A41
 91B16
 91B30
 91B70
 Industry Sectors
 Authors

 TieXin Guo ^{(1)}
 Author Affiliations

 1. LMIB and School of Mathematics and Systems Science, Beihang University, Beijing, 100191, China