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

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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 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 L⁰-convex topology and in particular a characterization for a locally L⁰-convex module to be L⁰-pre-barreled. Section 7 gives some basic results on L⁰-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 Lᵖ-type of convex conditional risk measure (1≤ p<+∞) can be extended to an L∞ F(E)-type of σε,λ(L∞ F(E), L¹ F(E))-lower semicontinuous conditional convex risk measure and an Lᵖ F(E)-type of Tε,λ-continuous conditional convex risk measure (1≤ p<+∞), respectively.