, Volume 54, Issue 4, pp 633-660
Date: 24 Mar 2011

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

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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 -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.