The Annals of Applied Probability

Optimal stopping under adverse nonlinear expectation and related games

Marcel Nutz and Jianfeng Zhang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study the existence of optimal actions in a zero-sum game $\inf_{\tau}\sup_{P}E^{P}[X_{\tau}]$ between a stopper and a controller choosing a probability measure. This includes the optimal stopping problem $\inf_{\tau}\mathcal{E}(X_{\tau})$ for a class of sublinear expectations $\mathcal{E}(\cdot)$ such as the $G$-expectation. We show that the game has a value. Moreover, exploiting the theory of sublinear expectations, we define a nonlinear Snell envelope $Y$ and prove that the first hitting time $\inf\{t:Y_{t}=X_{t}\}$ is an optimal stopping time. The existence of a saddle point is shown under a compactness condition. Finally, the results are applied to the subhedging of American options under volatility uncertainty.

Article information

Ann. Appl. Probab., Volume 25, Number 5 (2015), 2503-2534.

Received: August 2013
Revised: June 2014
First available in Project Euclid: 30 July 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 93E20: Optimal stochastic control 49L20: Dynamic programming method 91A15: Stochastic games 60G44: Martingales with continuous parameter 91B28

Controller-and-stopper game optimal stopping saddle point nonlinear expectation $G$-expectation


Nutz, Marcel; Zhang, Jianfeng. Optimal stopping under adverse nonlinear expectation and related games. Ann. Appl. Probab. 25 (2015), no. 5, 2503--2534. doi:10.1214/14-AAP1054.

Export citation


  • [1] Avellaneda, M., Levy, A. and Parás, A. (1995). Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance 2 73–88.
  • [2] Bayraktar, E. and Huang, Y.-J. (2013). On the multidimensional controller-and-stopper games. SIAM J. Control Optim. 51 1263–1297.
  • [3] Bayraktar, E., Karatzas, I. and Yao, S. (2010). Optimal stopping for dynamic convex risk measures. Illinois J. Math. 54 1025–1067 (2012).
  • [4] Bayraktar, E. and Yao, S. Robust optimal stopping under volatility uncertainty. Preprint. Available at arXiv:1301.0091v5.
  • [5] Bayraktar, E. and Yao, S. (2011). Optimal stopping for non-linear expectations—Part I. Stochastic Process. Appl. 121 185–211.
  • [6] Bayraktar, E. and Yao, S. (2011). Optimal stopping for non-linear expectations—Part II. Stochastic Process. Appl. 121 212–264.
  • [7] Bertsekas, D. P. and Shreve, S. E. (1978). Stochastic Optimal Control: The Discrete Time Case. Mathematics in Science and Engineering 139. Academic Press, New York.
  • [8] Cheridito, P., Soner, H. M., Touzi, N. and Victoir, N. (2007). Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Comm. Pure Appl. Math. 60 1081–1110.
  • [9] Ekren, I., Touzi, N. and Zhang, J. (2014). Optimal stopping under nonlinear expectation. Stochastic Process. Appl. 124 3277–3311.
  • [10] El Karoui, N. (1981). Les aspects probabilistes du contrôle stochastique. In Ninth Saint Flour Probability Summer School—1979 (Saint Flour, 1979). Lecture Notes in Math. 876 73–238. Springer, Berlin.
  • [11] El Karoui, N., Pardoux, E. and Quenez, M. C. (1997). Reflected backward SDEs and American options. In Numerical Methods in Finance. Publ. Newton Inst. 13 215–231. Cambridge Univ. Press, Cambridge.
  • [12] Fleming, W. H. and Souganidis, P. E. (1989). On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38 293–314.
  • [13] Föllmer, H. and Schied, A. (2004). Stochastic Finance: An Introduction in Discrete Time, extended ed. De Gruyter Studies in Mathematics 27. de Gruyter, Berlin.
  • [14] Karandikar, R. L. (1995). On pathwise stochastic integration. Stochastic Process. Appl. 57 11–18.
  • [15] Karatzas, I. and Sudderth, W. (2006). Stochastic games of control and stopping for a linear diffusion. In Random Walk, Sequential Analysis and Related Topics: A Festschrift in Honor of Y. S. Chow (A. Hsiung, Zh. Ying and C. H. Zhang, eds.) 100–117. World Sci. Publ., Hackensack, NJ.
  • [16] Karatzas, I. and Sudderth, W. D. (2001). The controller-and-stopper game for a linear diffusion. Ann. Probab. 29 1111–1127.
  • [17] Karatzas, I. and Zamfirescu, I.-M. (2005). Game approach to the optimal stopping problem. Stochastics 77 401–435.
  • [18] Karatzas, I. and Zamfirescu, I.-M. (2008). Martingale approach to stochastic differential games of control and stopping. Ann. Probab. 36 1495–1527.
  • [19] Lyons, T. J. (1995). Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance 2 117–133.
  • [20] Maitra, A. P. and Sudderth, W. D. (1996). The gambler and the stopper. In Statistics, Probability and Game Theory. Institute of Mathematical Statistics Lecture Notes—Monograph Series 30 191–208. IMS, Hayward, CA.
  • [21] Matoussi, A., Piozin, L. and Possamaï, D. (2014). Second-order BSDEs with general reflection and game options under uncertainty. Stochastic Process. Appl. 124 2281–2321.
  • [22] Matoussi, A., Possamai, D. and Zhou, C. (2013). Second order reflected backward stochastic differential equations. Ann. Appl. Probab. 23 2420–2457.
  • [23] Neufeld, A. and Nutz, M. (2013). Superreplication under volatility uncertainty for measurable claims. Electron. J. Probab. 18 1–14.
  • [24] Neufeld, A. and Nutz, M. (2014). Nonlinear Lévy processes and their characteristics. Preprint. Available at arXiv:1401.7253v1.
  • [25] Nutz, M. (2012). A quasi-sure approach to the control of non-Markovian stochastic differential equations. Electron. J. Probab. 17 1–23.
  • [26] Nutz, M. and van Handel, R. (2013). Constructing sublinear expectations on path space. Stochastic Process. Appl. 123 3100–3121.
  • [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] Peng, S. (2010). Backward stochastic differential equation, nonlinear expectation and their applications. In Proceedings of the International Congress of Mathematicians. Volume I (R. Bhatia, ed.) 393–432. Hindustan Book Agency, New Delhi.
  • [30] Riedel, F. (2009). Optimal stopping with multiple priors. Econometrica 77 857–908.
  • [31] Smith, A. T. (2002). American options under uncertain volatility. Appl. Math. Finance 9 123–141.
  • [32] Soner, H. M., Touzi, N. and Zhang, J. (2012). Wellposedness of second order backward SDEs. Probab. Theory Related Fields 153 149–190.
  • [33] Soner, H. M., Touzi, N. and Zhang, J. (2013). Dual formulation of second order target problems. Ann. Appl. Probab. 23 308–347.
  • [34] Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Springer, New York.
  • [35] Zheng, W. A. (1985). Tightness results for laws of diffusion processes application to stochastic mechanics. Ann. Inst. Henri Poincaré Probab. Stat. 21 103–124.