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

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• Tiexin Guo

## 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 $L^{\infty}-$type of conditional convex risk measure and every continuous $L^{p}-$type of convex conditional risk measure (\$1\leq p

## Suggested Citation

• Tiexin Guo, 2010. "Recent progress in random metric theory and its applications to conditional risk measures," Papers 1006.0697, arXiv.org, revised Mar 2011.
• Handle: RePEc:arx:papers:1006.0697
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File URL: http://arxiv.org/pdf/1006.0697

## References listed on IDEAS

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1. Rosazza Gianin, Emanuela, 2006. "Risk measures via g-expectations," Insurance: Mathematics and Economics, Elsevier, vol. 39(1), pages 19-34, August.
2. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
3. Andrzej Ruszczynski & Alexander Shapiro, 2004. "Optimization of Convex Risk Functions," Risk and Insurance 0404001, University Library of Munich, Germany, revised 08 Oct 2005.
4. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
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## Citations

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Cited by:

1. Tiexin Guo & Shien Zhao & Xiaolin Zeng, 2012. "On random convex analysis -- the analytic foundation of the module approach to conditional risk measures," Papers 1210.1848, arXiv.org, revised Mar 2013.
2. Beatrice Acciaio & Verena Goldammer, 2013. "Optimal portfolio selection via conditional convex risk measures on L p," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 36(1), pages 1-21, May.

### NEP fields

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