Electronic Journal of Probability

Superreplication under volatility uncertainty for measurable claims

Ariel Neufeld and Marcel Nutz

Full-text: Open access


We establish the duality-formula for the superreplication price in a setting of volatility uncertainty which includes the example of "random $G$-expectation". In contrast to previous results, the contingent claim is not assumed to be quasi-continuous.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 48, 14 pp.

Accepted: 15 April 2013
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 93E20: Optimal stochastic control
Secondary: 91B30: Risk theory, insurance 91B28

Volatility uncertainty Superreplication Nonlinear expectation

This work is licensed under a Creative Commons Attribution 3.0 License.


Neufeld, Ariel; Nutz, Marcel. Superreplication under volatility uncertainty for measurable claims. Electron. J. Probab. 18 (2013), paper no. 48, 14 pp. doi:10.1214/EJP.v18-2358. https://projecteuclid.org/euclid.ejp/1465064273

Export citation


  • Bertsekas, Dimitri P.; Shreve, Steven E. Stochastic optimal control. The discrete time case. Mathematics in Science and Engineering, 139. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. xiii+323 pp. ISBN: 0-12-093260-1
  • Bichteler, Klaus. Stochastic integration and $L^{p}$-theory of semimartingales. Ann. Probab. 9 (1981), no. 1, 49–89.
  • Chung, Dong M. On the local convergence in measure. Bull. Korean Math. Soc. 18 (1981/82), no. 2, 51–53.
  • Dellacherie, Claude; Meyer, Paul-André. Probabilities and potential. North-Holland Mathematics Studies, 29. North-Holland Publishing Co., Amsterdam-New York; North-Holland Publishing Co., Amsterdam-New York, 1978. viii+189 pp. ISBN: 0-7204-0701-X
  • Dellacherie, Claude; Meyer, Paul-André. Probabilities and potential. B. Theory of martingales. Translated from the French by J. P. Wilson. North-Holland Mathematics Studies, 72. North-Holland Publishing Co., Amsterdam, 1982. xvii+463 pp. ISBN: 0-444-86526-8
  • Denis, Laurent; Martini, Claude. A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16 (2006), no. 2, 827–852.
  • Doob, J. L. Measure theory. Graduate Texts in Mathematics, 143. Springer-Verlag, New York, 1994. xii+210 pp. ISBN: 0-387-94055-3
  • Jacod, Jean; Yor, Marc. Étude des solutions extrémales et représentation intégrale des solutions pour certains problèmes de martingales. (French) Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 38 (1977), no. 2, 83–125.
  • Nutz, Marcel. Random G-expectations. To appear in Ann. Appl. Probab., 2010.
  • Nutz, Marcel. Pathwise construction of stochastic integrals. Electron. Commun. Probab. 17 (2012), no. 24, 7 pp.
  • Nutz, Marcel. A quasi-sure approach to the control of non-Markovian stochastic differential equations. Electron. J. Probab. 17 (2012), no. 23, 23 pp.
  • Nutz, Marcel; Soner, H. Mete. Superhedging and dynamic risk measures under volatility uncertainty. SIAM J. Control Optim. 50 (2012), no. 4, 2065–2089.
  • M. Nutz and R. van Handel. Constructing sublinear expectations on path space. To appear in Stochastic Process. Appl., 2012.
  • Nutz, M. and Zhang, J. Optimal stopping under adverse nonlinear expectation and related games. Preprint arXiv:1212.2140v1, 2012.
  • Peng, Shige. $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô type. Stochastic analysis and applications, 541–567, Abel Symp., 2, Springer, Berlin, 2007.
  • Peng, Shige. Multi-dimensional $G$-Brownian motion and related stochastic calculus under $G$-expectation. Stochastic Process. Appl. 118 (2008), no. 12, 2223–2253.
  • S. Peng. Nonlinear expectations and stochastic calculus under uncertainty. Preprint arXiv:1002.4546v1, 2010.
  • D. Possamai, G. Royer, and N. Touzi. On the robust superhedging of measurable claims. Preprint arXiv:1302.1850v1, 2013.
  • Protter, Philip E. Stochastic integration and differential equations. Second edition. Version 2.1. Corrected third printing. Stochastic Modelling and Applied Probability, 21. Springer-Verlag, Berlin, 2005. xiv+419 pp. ISBN: 3-540-00313-4
  • Soner, H. Mete; Touzi, Nizar. Dynamic programming for stochastic target problems and geometric flows. J. Eur. Math. Soc. (JEMS) 4 (2002), no. 3, 201–236.
  • Soner, H. Mete; Touzi, Nizar; Zhang, Jianfeng. Martingale representation theorem for the $G$-expectation. Stochastic Process. Appl. 121 (2011), no. 2, 265–287.
  • Soner, H. Mete; Touzi, Nizar; Zhang, Jianfeng. Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16 (2011), no. 67, 1844–1879.
  • H. M. Soner, N. Touzi, and J. Zhang. Dual formulation of second order target problems. Ann. Appl. Probab., 23(1):308–347, 2013.
  • Song, YongSheng. Some properties on $G$-evaluation and its applications to $G$-martingale decomposition. Sci. China Math. 54 (2011), no. 2, 287–300.
  • Stroock, Daniel W.; Varadhan, S. R. Srinivasa. Multidimensional diffusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 233. Springer-Verlag, Berlin-New York, 1979. xii+338 pp. ISBN: 3-540-90353-4