The Annals of Applied Probability

Optimal stopping under adverse nonlinear expectation and related games

Marcel Nutz and Jianfeng Zhang

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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

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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.

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  • [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.