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On random convex analysis -- the analytic foundation of the module approach to conditional risk measures

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  • Tiexin Guo
  • Shien Zhao
  • Xiaolin Zeng

Abstract

To provide a solid analytic foundation for the module approach to conditional risk measures, this paper establishes a complete random convex analysis over random locally convex modules by simultaneously considering the two kinds of topologies (namely the $(\varepsilon,\lambda)$--topology and the locally $L^0$-- convex topology). Then, we make use of the advantage of the $(\varepsilon,\lambda)$--topology and grasp the local property of $L^0$--convex conditional risk measures to prove that every $L^{0}$--convex $L^{p}$--conditional risk measure ($1\leq p\leq+\infty$) can be uniquely extended to an $L^{0}$--convex $L^{p}_{\mathcal{F}}(\mathcal{E})$--conditional risk measure and that the dual representation theorem of the former can also be regarded as a special case of that of the latter, which shows that the study of $L^p$--conditional risk measures can be incorporated into that of $L^{p}_{\mathcal{F}}(\mathcal{E})$--conditional risk measures. In particular, in the process we find that combining the countable concatenation hull of a set and the local property of conditional risk measures is a very useful analytic skill that may considerably simplify and improve the study of $L^{0}$--convex conditional risk measures.

Suggested Citation

  • 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.
  • Handle: RePEc:arx:papers:1210.1848
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    References listed on IDEAS

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    1. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and dynamic convex risk measures," Finance and Stochastics, Springer, vol. 9(4), pages 539-561, October.
    2. Freddy Delbaen & Shige Peng & Emanuela Rosazza Gianin, 2010. "Representation of the penalty term of dynamic concave utilities," Finance and Stochastics, Springer, vol. 14(3), pages 449-472, September.
    3. Tiexin Guo, 2010. "Recent progress in random metric theory and its applications to conditional risk measures," Papers 1006.0697, arXiv.org, revised Mar 2011.
    4. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and Dynamic Convex Risk Measures," SFB 649 Discussion Papers SFB649DP2005-006, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
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