The Annals of Applied Probability

Risk measuring under model uncertainty

Jocelyne Bion-Nadal and Magali Kervarec

Full-text: Open access


The framework of this paper is that of risk measuring under uncertainty which is when no reference probability measure is given. To every regular convex risk measure on $\mathcal{C}_{b}(\Omega)$, we associate a unique equivalence class of probability measures on Borel sets, characterizing the riskless nonpositive elements of $\mathcal{C}_{b}(\Omega)$. We prove that the convex risk measure has a dual representation with a countable set of probability measures absolutely continuous with respect to a certain probability measure in this class. To get these results we study the topological properties of the dual of the Banach space L1(c) associated to a capacity c.

As application, we obtain that every G-expectation $\mathbb{E}$ has a representation with a countable set of probability measures absolutely continuous with respect to a probability measure P such that P(|f|) = 0 if and only iff $\mathbb{E}(|f|)=0$. We also apply our results to the case of uncertain volatility.

Article information

Ann. Appl. Probab. Volume 22, Number 1 (2012), 213-238.

First available in Project Euclid: 7 February 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46A20: Duality theory 91B30: Risk theory, insurance
Secondary: 46E05: Lattices of continuous, differentiable or analytic functions

Risk measure duality theory uncertainty capacity


Bion-Nadal, Jocelyne; Kervarec, Magali. Risk measuring under model uncertainty. Ann. Appl. Probab. 22 (2012), no. 1, 213--238. doi:10.1214/11-AAP766.

Export citation


  • [1] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9 203–228.
  • [2] Avellaneda, M., Levy, A. and Paras, A. (1995). Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance 2 73–88.
  • [3] Barrieu, P. and El Karoui, N. (2009). Pricing, hedging and optimally designing derivatives via minimization of risk measures. In Volume on Indifference Pricing (R. Carmona, ed.). Princeton Univ. Press, Princeton, NJ.
  • [4] Biagini, S. and Frittelli, M. (2009). On the extension of the Namioka–Klee theorem and on the Fatou property for risk measures. In Optimality and Risk—Modern Trends in Mathematical Finance: The Kabanov Festschrift (F. Delbaen, M. Rásonyi and C. Stricker, eds.) 1–28. Springer, Berlin.
  • [5] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [6] Bion-Nadal, J. (2004). Conditional risk measure and robust representation of convex conditional risk measures. Preprint CMAP 557.
  • [7] Bion-Nadal, J. (2008). Dynamic risk measures: Time consistency and risk measures from BMO martingales. Finance Stoch. 12 219–244.
  • [8] Bion-Nadal, J. (2009). Time consistent dynamic risk processes. Stochastic Process. Appl. 119 633–654.
  • [9] Bourbaki, N. (1958). Eléments de Mathématiques, Topologie Générale, Chapter X, 2nd ed. Hermann, Paris.
  • [10] Bourbaki, N. (1969). Eléments de Mathématiques, Integration, Chapter 9. Hermann.
  • [11] Cheridito, P., Delbaen, F. and Kupper, M. (2006). Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Probab. 11 57–106.
  • [12] Delbaen, F. (2002). Coherent risk measures on general probability spaces. In Advances in Finance and Stochastics. Essays in Honour of Dieter Sondermann (K. Sandmann and P. J. Schönbucher, eds.) 1–37. Springer, Berlin.
  • [13] Delbaen, F. (2006). The structure of m-stable sets and in particular of the set of risk neutral measures. In In Memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX. Lecture Notes in Math. 1874 215–258. Springer, Berlin.
  • [14] Delbaen, F., Peng, S. and Rosazza Gianin, E. (2010). Representation of the penalty term of dynamic concave utilities. Finance Stoch. 14 449–472.
  • [15] Denis, L., Hu, M. and Peng, S. (2011). Function spaces and capacity related to a sublinear expectation: Application to G-Brownian motion pathes. Potential Anal. 34 139–161.
  • [16] Denis, L. and Martini, C. (2006). A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16 827–852.
  • [17] Detlefsen, K. and Scandolo, G. (2005). Conditional and dynamic convex risk measures. Finance Stoch. 9 539–561.
  • [18] Dunford, N. and Schwartz, J. T. (1958). Linear Operators. Part I. General Theory. Wiley, New York.
  • [19] Feyel, D. and de la Pradelle, A. (1977). Topologies fines et compactifications associées à certains espaces de Dirichlet. Ann. Inst. Fourier (Grenoble) 27 121–146.
  • [20] Feyel, D. and de La Pradelle, A. (1989). Espaces de Sobolev Gaussiens. Ann. Inst. Fourier (Grenoble) 39 875–908.
  • [21] Föllmer, H. and Schied, A. (2004). Stochastic Finance: An Introduction in Discrete Time, 2nd ed. de Gruyter Studies in Mathematics 27. de Gruyter, Berlin.
  • [22] Frittelli, M. and Rosazza Gianin, E. (2002). Putting order in risk measures. Journal of Banking Finance 26 1473–1486.
  • [23] Hu, M.-S. and Peng, S.-G. (2009). On representation theorem of G-expectations and paths of G-Brownian motion. Acta Math. Appl. Sin. Engl. Ser. 25 539–546.
  • [24] Kervarec, M. (2008). Modèles non dominés en mathématiques financières. Thèse de Doctorat en Mathématiques, Univ. d’Evry, Évry, France.
  • [25] Klöppel, S. and Schweizer, M. (2007). Dynamic indifference valuation via convex risk measures. Math. Finance 17 599–627.
  • [26] Peng, S. (2004). Nonlinear expectations, nonlinear evaluations and risk measures. In Stochastic Methods in Finance. Lecture Notes in Math. 1856 165–253. Springer, Berlin.
  • [27] Peng, S. (2007). G-expectation, G-Brownian motion and related stochastic calculus of Itô type. In Stochastic Analysis and Applications. Abel Symp. 2 541–567. Springer, Berlin.
  • [28] Peng, S. (2008). Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stochastic Process. Appl. 118 2223–2253.
  • [29] Rockafellar, R. T. (1974). Conjugate Duality and Optimization. SIAM, Philadelphia, PA.
  • [30] Roorda, B. and Schumacher, J. M. (2007). Time consistency conditions for acceptability measures, with an application to Tail Value at Risk. Insurance Math. Econom. 40 209–230.
  • [31] Rudin, W. (1966). Real and Complex Analysis. McGraw-Hill, New York.