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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.

### MSC:

46S50 | Functional analysis in probabilistic metric linear spaces |

46A22 | Theorems of Hahn-Banach type; extension and lifting of functionals and operators |

46A25 | Reflexivity and semi-reflexivity |

46H25 | Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) |

47H40 | Random nonlinear operators |

52A41 | Convex functions and convex programs in convex geometry |

91B16 | Utility theory |

91B30 | Risk theory, insurance (MSC2010) |

91B70 | Stochastic models in economics |

### Keywords:

random normed module; random inner product module; random locally convex module; random conjugate space; \(L^{0}\)-convex analysis; conditional risk measures### References:

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