Risk measuring under model uncertainty

Annals of Applied Probability

Volumn 22, Issue 1, 2012, Pages 213-238

  Bion Nadal, Jocelyne ^a   Kervarec, Magali ^b  

^a CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

^b UNIVERSITÉ D EVRY VAL D ESSONNE   (France)

Author keywords

Capacity; Duality theory; Risk measure; Uncertainty

Indexed keywords

EID: 84866432110     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/11-AAP766     Document Type: Article

References (31)

1. ARTZNER, P., DELBAEN, F., EBER, J.-M. and HEATH, D. (1999). Coherent measures of risk. Math. Finance 9 203-228. MR1850791
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. MR2648595
5. BILLINGSLEY, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York. MR1700749
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. MR2390189
8. BION-NADAL, J. (2009). Time consistent dynamic risk processes. Stochastic Process. Appl. 119 633-654. MR2494007
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. MR2199055 (Pubitemid 43157373)
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. MR1929369
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. MR2276899 (Pubitemid 44597593)
14. DELBAEN, F., PENG, S. and ROSAZZA GIANIN, E. (2010). Representation of the penalty term of dynamic concave utilities. Finance Stoch. 14 449-472. MR2670421
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. MR2244434 (Pubitemid 44196534)
17. DETLEFSEN, K. and SCANDOLO, G. (2005). Conditional and dynamic convex risk measures. Finance Stoch. 9 539-561. MR2212894 (Pubitemid 41232084)
18. DUNFORD, N. and SCHWARTZ, J. T. (1958). Linear Operators. Part I. General Theory.Wiley, New York. MR1009162
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. MR0508018
20. FEYEL, D. and DE LA PRADELLE, A. (1989). Espaces de Sobolev Gaussiens. Ann. Inst. Fourier (Grenoble) 39 875-908. MR1036336
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. MR2169807
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. MR2506990
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. MR2352907
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. MR2113723
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. MR2397805
28. PENG, S. (2008). Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stochastic Process. Appl. 118 2223-2253. MR2474349
29. ROCKAFELLAR, R. T. (1974). Conjugate Duality and Optimization. SIAM, Philadelphia, PA. MR0373611
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. MR2286942
31. RUDIN, W. (1966). Real and Complex Analysis. McGraw-Hill, New York. MR0210528