Science China Mathematics

, Volume 54, Issue 4, pp 633–660

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


DOI: 10.1007/s11425-011-4189-6

Cite this article as:
Guo, T. Sci. China Math. (2011) 54: 633. doi:10.1007/s11425-011-4189-6


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 \( L_\mathcal{F}^\infty \left( \mathcal{E} \right) \)-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.


random normed module random inner product module random locally convex module random conjugate space L0-convex analysis conditional risk measures 


46A22 46A25 46H25 47H40 52A41 91B16 91B30 91B70 

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.LMIB and School of Mathematics and Systems ScienceBeihang UniversityBeijingChina