Top

Published in:

2018 | OriginalPaper | Chapter

Conditional Submodular Coherent Risk Measures

Authors : Giulianella Coletti, Davide Petturiti, Barbara Vantaggi

Published in: Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Web Apps and Imprecise Probabilitites

next chapter On the Structure of Group-Like FL-chains

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). https://doi.org/10.1007/s10479-006-0132-6MathSciNetCrossRefMATH

Artzner, P., Delbaen, P., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finan. 9(3), 203–228 (1999). https://doi.org/10.1111/1467-9965.00068MathSciNetCrossRefMATH

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). https://doi.org/10.1142/S0218488509005796MathSciNetCrossRefMATH

Bion-Nadal, J.: Dynamic risk measures: time consistency and risk measures from BMO martingales. Finan. Stochast. 12, 219–244 (2008). https://doi.org/10.1007/s00780-007-0057-1MathSciNetCrossRefMATH

Capotorti, A., Coletti, G., Vantaggi, B.: Standard and nonstandard representability of positive uncertainty orderings. Kybernetika 50(2), 189–215 (2014). https://doi.org/10.14736/kyb-2014-2-0189MathSciNetCrossRefMATH

Chateauneuf, A., Kast, R., Lapied, A.: Conditioning capacities and Choquet integrals: the role of comonotony. Theor. Dec. 51, 367–386 (2001). https://doi.org/10.1023/A:1015567329595MathSciNetCrossRefMATH

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). https://doi.org/10.1016/0165-4896(89)90056-5CrossRefMATH

Cheridito, P., Delbaen, F., Kupper, M.: Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Probab. 11, 57–106 (2006). https://doi.org/10.1214/EJP.v11-302MathSciNetCrossRefMATH

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). https://doi.org/10.1016/j.ins.2015.12.020MathSciNetCrossRefMATH

10.

Coletti, G., Vantaggi, B.: A view on conditional measures through local representability of binary relations. Int. J. Approx. Reason. 47, 268–283 (2008). https://doi.org/10.1016/j.ijar.2007.05.007MathSciNetCrossRefMATH

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). https://doi.org/10.1007/978-3-662-04790-3_1CrossRef

12.

Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat. 2, 325–339 (1967). https://doi.org/10.1214/aoms/1177698950MathSciNetCrossRefMATH

13.

Denneberg, D.: Conditioning (updating) non-additive measures. Ann. Oper. Res. 52(1), 21–42 (1994). https://doi.org/10.1007/BF02032159MathSciNetCrossRefMATH

14.

Denneberg, D.: Non-additive Measure and Integral. Kluwer Academic Publishers, Dordrecht (1994). https://doi.org/10.1007/978-94-017-2434-0CrossRefMATH

15.

Dubins, L.E.: Finitely additive conditional probabilities, conglomerability and disintegrations. Ann. Probab. 3, 89–99 (1975). https://doi.org/10.1214/aop/1176996451MathSciNetCrossRefMATH

16.

Föllmer, H., Penner, I.: Convex risk measures and the dynamics of their penalty functions. Stat. Dec. 24, 61–96 (2006). https://doi.org/10.1524/stnd.2006.24.1.61MathSciNetCrossRefMATH

17.

Grabisch, M.: Set Functions, Games and Capacities in Decision Making. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-30690-2CrossRefMATH

18.

Halpern, J.H.: Reasoning About Uncertainty. The MIT Press, Cambrige (2005)MATH

19.

Jaffray, J.Y.: Bayesian updating and belief functions. IEEE Trans. Man Cybern. 22, 1144–1152 (1992). https://doi.org/10.1109/21.179852MathSciNetCrossRefMATH

20.

Pelessoni, R., Vicig, P.: Imprecise previsions for risk measurement. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 11(4), 393–412 (2003). https://doi.org/10.1142/S0218488503002156MathSciNetCrossRefMATH

21.

Pelessoni, R., Vicig, P.: Uncertainty modelling and conditioning with convex imprecise previsions. Int. J. Approx. Reason. 39, 297–319 (2005). https://doi.org/10.1016/j.ijar.2004.10.007MathSciNetCrossRefMATH

22.

Pelessoni, R., Vicig, P.: 2-coherent and 2-convex conditional lower previsions. Int. J. Approx. Reason. 77, 66–86 (2016). https://doi.org/10.1016/j.ijar.2016.06.003MathSciNetCrossRefMATH

23.

Pelessoni, R., Vicig, P., Zaffalon, M.: Inference and risk measurement with the pari-mutuel model. Int. J. Approx. Reason. 51, 1145–1158 (2010). https://doi.org/10.1016/j.ijar.2010.08.005MathSciNetCrossRefMATH

24.

Riedel, F.: Dynamic coherent risk measures. Stochast. Process. Appl. 112(2), 185–200 (2004). https://doi.org/10.1016/j.spa.2004.03.004MathSciNetCrossRefMATH

25.

Roorda, B., Schumacher, J.M., Engwerda, J.: Coherent acceptability measures in multiperiod models. Math. Finan. 15(4), 589–612 (2005). https://doi.org/10.1111/j.1467-9965.2005.00252.xMathSciNetCrossRefMATH

26.

Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 97(2), 255–261 (1986). https://doi.org/10.1090/S0002-9939-1986-0835875-8MathSciNetCrossRefMATH

27.

Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)MATH

28.

Troffaes, M.C.M., de Cooman, G.: Lower Previsions. Wiley, Hoboken (2014). https://doi.org/10.1002/9781118762622CrossRefMATH

29.

Vicig, P.: Financial risk measurement with imprecise probabilities. Int. J. Approx. Reason. 49, 159–174 (2008). https://doi.org/10.1016/j.ijar.2007.06.009MathSciNetCrossRefMATH

30.

Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, New York (1991)CrossRef

Title: Conditional Submodular Coherent Risk Measures
Authors: Giulianella Coletti
Davide Petturiti
Barbara Vantaggi
Publisher: Springer International Publishing
Book: Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations
Print ISBN: 978-3-319-91475-6

Electronic ISBN: 978-3-319-91476-3

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-91476-3_20

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner