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

By T. Guo. 2010

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AbstractThe 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 $$ L_\mathcal{F}^\infty \left( \mathcal{E} \right) $$ -type of σε,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.

Subfields: Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/397e5bfb3654a02283958a452eb40d467b8c2a6b