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 ^∞-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.