Conditional Submodular Coherent Risk Measures

  • Giulianella Coletti
  • Davide PetturitiEmail author
  • Barbara Vantaggi
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 854)


A family of conditional risk measures is introduced by considering a single period financial market, relying on a notion of conditioning for submodular capacities, which generalizes that introduced by Dempster. The resulting measures are expressed as discounted conditional Choquet expected values, take into account ambiguity towards uncertainty and allow for conditioning to “null” events. We also provide a characterisation of consistence of a partial assessment with a conditional submodular coherent risk measure. The latter amounts to test the solvability of a suitable sequence of linear systems.


Coherent risk measure Conditional Choquet expected value Conditional submodular capacity 


  1. 1.
    Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., Ku, H.: Coherent multiperiod risk adjusted values and Bellman’s principle. Ann. Oper. Res. 152, 5–22 (2007). Scholar
  2. 2.
    Artzner, P., Delbaen, P., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finan. 9(3), 203–228 (1999). Scholar
  3. 3.
    Baroni, P., Pelessoni, R., Vicig, P.: Generalizing Dutch risk measures through imprecise previsions. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 17(2), 153–177 (2009). Scholar
  4. 4.
    Bion-Nadal, J.: Dynamic risk measures: time consistency and risk measures from BMO martingales. Finan. Stochast. 12, 219–244 (2008). Scholar
  5. 5.
    Capotorti, A., Coletti, G., Vantaggi, B.: Standard and nonstandard representability of positive uncertainty orderings. Kybernetika 50(2), 189–215 (2014). Scholar
  6. 6.
    Chateauneuf, A., Kast, R., Lapied, A.: Conditioning capacities and Choquet integrals: the role of comonotony. Theor. Dec. 51, 367–386 (2001). Scholar
  7. 7.
    Chateauneuf, A., Jaffray, J.-Y.: Some characterizations of lower probabilities and other monotone capacities through the use of Möbous inversion. Math. Soc. Sci. 17, 263–283 (1989). Scholar
  8. 8.
    Cheridito, P., Delbaen, F., Kupper, M.: Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Probab. 11, 57–106 (2006). Scholar
  9. 9.
    Coletti, G., Petturiti, D., Vantaggi, B.: Conditional belief functions as lower envelopes of conditional probabilities in a finite setting. Inf. Sci. 339, 64–84 (2016). Scholar
  10. 10.
    Coletti, G., Vantaggi, B.: A view on conditional measures through local representability of binary relations. Int. J. Approx. Reason. 47, 268–283 (2008). Scholar
  11. 11.
    Delbaen, F.: Coherent risk measures on general probability spaces. In: Sandman, K., Schönbucher, P.J. (eds.) Advances in Finance Stochastics, pp. 1–37. Springer, Heidelberg (2002). Scholar
  12. 12.
    Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat. 2, 325–339 (1967). Scholar
  13. 13.
    Denneberg, D.: Conditioning (updating) non-additive measures. Ann. Oper. Res. 52(1), 21–42 (1994). Scholar
  14. 14.
    Denneberg, D.: Non-additive Measure and Integral. Kluwer Academic Publishers, Dordrecht (1994). Scholar
  15. 15.
    Dubins, L.E.: Finitely additive conditional probabilities, conglomerability and disintegrations. Ann. Probab. 3, 89–99 (1975). Scholar
  16. 16.
    Föllmer, H., Penner, I.: Convex risk measures and the dynamics of their penalty functions. Stat. Dec. 24, 61–96 (2006). Scholar
  17. 17.
    Grabisch, M.: Set Functions, Games and Capacities in Decision Making. Springer, Cham (2016). Scholar
  18. 18.
    Halpern, J.H.: Reasoning About Uncertainty. The MIT Press, Cambrige (2005)zbMATHGoogle Scholar
  19. 19.
    Jaffray, J.Y.: Bayesian updating and belief functions. IEEE Trans. Man Cybern. 22, 1144–1152 (1992). Scholar
  20. 20.
    Pelessoni, R., Vicig, P.: Imprecise previsions for risk measurement. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 11(4), 393–412 (2003). Scholar
  21. 21.
    Pelessoni, R., Vicig, P.: Uncertainty modelling and conditioning with convex imprecise previsions. Int. J. Approx. Reason. 39, 297–319 (2005). Scholar
  22. 22.
    Pelessoni, R., Vicig, P.: 2-coherent and 2-convex conditional lower previsions. Int. J. Approx. Reason. 77, 66–86 (2016). Scholar
  23. 23.
    Pelessoni, R., Vicig, P., Zaffalon, M.: Inference and risk measurement with the pari-mutuel model. Int. J. Approx. Reason. 51, 1145–1158 (2010). Scholar
  24. 24.
    Riedel, F.: Dynamic coherent risk measures. Stochast. Process. Appl. 112(2), 185–200 (2004). Scholar
  25. 25.
    Roorda, B., Schumacher, J.M., Engwerda, J.: Coherent acceptability measures in multiperiod models. Math. Finan. 15(4), 589–612 (2005). Scholar
  26. 26.
    Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 97(2), 255–261 (1986). Scholar
  27. 27.
    Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)zbMATHGoogle Scholar
  28. 28.
    Troffaes, M.C.M., de Cooman, G.: Lower Previsions. Wiley, Hoboken (2014). Scholar
  29. 29.
    Vicig, P.: Financial risk measurement with imprecise probabilities. Int. J. Approx. Reason. 49, 159–174 (2008). Scholar
  30. 30.
    Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, New York (1991)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Giulianella Coletti
    • 1
  • Davide Petturiti
    • 2
    Email author
  • Barbara Vantaggi
    • 3
  1. 1.Dip. Matematica e InformaticaUniversity of PerugiaPerugiaItaly
  2. 2.Dip. EconomiaUniversity of PerugiaPerugiaItaly
  3. 3.Dip. S.B.A.I.“La Sapienza” University of RomeRomeItaly

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