Skip to main content
Log in

The relations among the three kinds of conditional risk measures

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript


Let (Ω, E, P) be a probability space, F a sub-σ-algebra of E, L p(E) (1 ⩽ p ⩽ +∞) the classical function space and L p F (E) the L 0(F)-module generated by L p(E), which can be made into a random normed module in a natural way. Up to the present time, there are three kinds of conditional risk measures, whose model spaces are L (E), L p(E) (1 ⩽ p < +∞) and L p F (E) (1 ⩽ p ⩾ +∞) respectively, and a conditional convex dual representation theorem has been established for each kind. The purpose of this paper is to study the relations among the three kinds of conditional risk measures together with their representation theorems. We first establish the relation between L p(E) and L p F (E), namely L p F (E) = H cc (L p(E)), which shows that L p F (E) is exactly the countable concatenation hull of L p(E). Based on the precise relation, we then prove that every L 0(F)-convex L p(E)-conditional risk measure (1 ⩽ p ⩽ +∞) can be uniquely extended to an L 0(F)-convex L p F (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 F (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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Artzner P, Delbaen F, Eber J M, et al. Coherent measures of risk. Math Finance, 1999, 9: 203–228

    Article  MATH  MathSciNet  Google Scholar 

  2. Bion-Nadal J. Conditional risk measures and robust representation of convex conditional risk neasures. Preprint, 2004

    Google Scholar 

  3. Delbaen F. Coherent risk measures on general probability spaces. In: Sandmann K, Schönbucher P J, eds. Advances in Finance and Stochastics. Berlin: Springer, 2002, 1–37

    Chapter  Google Scholar 

  4. Delbaen F, Peng S G, Rosazza Gianin E. Representation of the penalty term of dynamic concave utilities. Finance Stoch, 2010, 14: 449–472

    Article  MATH  MathSciNet  Google Scholar 

  5. Detlefsen K, Scandolo G. Conditional and dynamic convex risk measures. Finance Stoch, 2005, 9: 539–561

    Article  MATH  MathSciNet  Google Scholar 

  6. Dunford N, Schwartz J T. Linear Operators (I). New York: Interscience, 1957

    Google Scholar 

  7. Ekeland I, Témam R. Convex Analysis and Variational Problems. New York: SIAM, 1999

    Book  MATH  Google Scholar 

  8. Filipović D, Kupper M, Vogelpoth N. Separation and duality in locally L0-convex modules. J Funct Anal, 2009, 256: 3996–4029

    Article  MATH  MathSciNet  Google Scholar 

  9. Filipović D, Kupper M, Vogelpoth N. Approaches to conditional risk. SIAM J Finance Math, 2012, 3: 402–432

    Article  MATH  Google Scholar 

  10. Filipović D, Svindland G. Convex risk measures beyond bounded risks, or the canonical model space for law-invariant convex risk measures is L 1. Working paper. Vienna: Vienna Institute of Finance, 2008

    Google Scholar 

  11. Föllmer H, Schied A. Stochastic Finance, An Introduction in Discrete Time, 2nd ed. Berlin: De Gruyter Stud Math, 2002

    Book  MATH  Google Scholar 

  12. Frittelli M, Rosazza Gianin E. Dynamic convex risk measures. In: Szegö G, ed. Risk Measures for the 21st Century. New York: Wiley, 2004, 227–248

    Google Scholar 

  13. Guo T X. Extension theorems of continuous random linear operators on random domains. J Math Anal Appl, 1995, 193: 15–27

    Article  MATH  MathSciNet  Google Scholar 

  14. Guo T X. A characterization for a complete random normed module to be random reflexive. J Xiamen Univ Natur Sci, 1997, 36: 167–170

    MATH  Google Scholar 

  15. Guo T X. Some basic theories of random normed linear spaces and random inner product spaces. Acta Anal Funct Appl, 1999, 1: 160–184

    MATH  MathSciNet  Google Scholar 

  16. Guo T X. Relations between some basic results derived from two kinds of topologies for a random locally convex module. J Funct Anal, 2010, 258: 3024–3047

    Article  MATH  MathSciNet  Google Scholar 

  17. Guo T X. Recent progress in random metric theory and its applications to conditional risk measures. Sci China Math, 2011, 54: 633–660

    Article  MATH  MathSciNet  Google Scholar 

  18. Guo T X, Chen X X. Random duality. Sci China Ser A, 2009, 52: 2084–2098

    Article  MATH  MathSciNet  Google Scholar 

  19. Guo T X, Li S B. The James theorem in complete random normed modules. J Math Anal Appl, 2005, 308: 257–265

    Article  MATH  MathSciNet  Google Scholar 

  20. Guo T X, Zhao S E, Zeng X L. On random convex analysis — the analytic foundation of the module approach to conditional risk measures. ArXiv:1210.1848

  21. Peng S G. Nonlinear expectations, nonlinear evaluations and risk measures. In: Lecture Notes in Mathematics, vol. 1856. New York: Springer, 2004, 165–253

    Google Scholar 

  22. Zowe J. A duality theorem for a convex programming problem in order complete lattices. J Math Anal Appl, 1975, 50: 273–287

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to TieXin Guo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, T., Zhao, S. & Zeng, X. The relations among the three kinds of conditional risk measures. Sci. China Math. 57, 1753–1764 (2014).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: