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Recent progress in random metric theory and its applications to conditional risk measures. (English) Zbl 1238.46058
Summary: 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 $L^{\infty}$-type of conditional convex risk measure and every continuous $L^{p}$-type of convex conditional risk measure $(1 \leq p < +\infty)$ can be extended to an $L_\mathcal{F}^\infty \left( \mathcal{E} \right)$-type of $\sigma_{\varepsilon ,\lambda} \left( {L_\mathcal{F}^\infty \left( \mathcal{E} \right),L_\mathcal{F}^1 \left( \mathcal{E} \right)} \right)$-lower semicontinuous conditional convex risk measure and an $L_\mathcal{F}^{p} \left( \mathcal{E} \right)$-type of $\mathcal{T}_{\varepsilon ,\lambda }$-continuous conditional convex risk measure $(1 \leq p < +\infty)$, respectively.

46S50Functional analysis in probabilistic metric linear spaces
46A22Theorems of Hahn-Banach type; extension and lifting of functionals and operators
46A25Reflexivity and semi-reflexivity of topological linear spaces
46H25Normed modules and Banach modules, topological modules
47H40Random operators (nonlinear)
52A41Convex functions and convex programs (convex geometry)
91B16Utility theory
91B30Risk theory, insurance
91B70Stochastic models in economics
Full Text: DOI
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