The Annals of Applied Probability

On the robust Dynkin game

Erhan Bayraktar and Song Yao

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 analyze a robust version of the Dynkin game over a set $\mathcal{P}$ of mutually singular probabilities. We first prove that conservative player’s lower and upper value coincide (let us denote the value by $V$). Such a result connects the robust Dynkin game with second-order doubly reflected backward stochastic differential equations. Also, we show that the value process $V$ is a submartingale under an appropriately defined nonlinear expectation $\underline{\mathscr{E}}$ up to the first time $\tau_{*}$ when $V$ meets the lower payoff process $L$. If the probability set $\mathcal{P}$ is weakly compact, one can even find an optimal triplet $(\mathbb{P}_{*},\tau_{*},\gamma_{*})$ for the value $V_{0}$.

The mutual singularity of probabilities in $\mathcal{P}$ causes major technical difficulties. To deal with them, we use some new methods including two approximations with respect to the set of stopping times.

Article information

Ann. Appl. Probab. Volume 27, Number 3 (2017), 1702-1755.

Received: July 2015
Revised: April 2016
First available in Project Euclid: 19 July 2017

Permanent link to this document

Digital Object Identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 93E20: Optimal stochastic control 49L20: Dynamic programming method 91A15: Stochastic games 91A55: Games of timing 60G44: Martingales with continuous parameter 91B28

Robust Dynkin game nonlinear expectation dynamic programming principle controls in weak formulation weak stability under pasting martingale approach path-dependent stochastic differential equations with controls optimal triplet optimal stopping with random maturity


Bayraktar, Erhan; Yao, Song. On the robust Dynkin game. Ann. Appl. Probab. 27 (2017), no. 3, 1702--1755. doi:10.1214/16-AAP1243.

Export citation


  • [1] Alario-Nazaret, M., Lepeltier, J.-P. and Marchal, B. (1982). Dynkin games. In Stochastic Differential Systems (Bad Honnef, 1982). Lect. Notes Control Inf. Sci. 43 23–32. Springer, Berlin.
  • [2] Alvarez, L. H. R. (2008). A class of solvable stopping games. Appl. Math. Optim. 58 291–314.
  • [3] Bayraktar, E., Karatzas, I. and Yao, S. (2010). Optimal stopping for dynamic convex risk measures. Illinois J. Math. 54 1025–1067.
  • [4] Bayraktar, E. and Sîrbu, M. (2014). Stochastic Perron’s method and verification without smoothness using viscosity comparison: Obstacle problems and Dynkin games. Proc. Amer. Math. Soc. 142 1399–1412.
  • [5] Bayraktar, E. and Yao, S. (2014). On the robust optimal stopping problem. SIAM J. Control Optim. 52 3135–3175.
  • [6] Bayraktar, E. and Yao, S. (2015). Doubly reflected BSDEs with integrable parameters and related Dynkin games. Stochastic Process. Appl. 125 4489–4542.
  • [7] Bayraktar, E. and Yao, S. (2015). Optimal stopping with random maturity under nonlinear expectations. Available at
  • [8] Bensoussan, A. and Friedman, A. (1974). Nonlinear variational inequalities and differential games with stopping times. J. Funct. Anal. 16 305–352.
  • [9] Bensoussan, A. and Friedman, A. (1977). Nonzero-sum stochastic differential games with stopping times and free boundary problems. Trans. Amer. Math. Soc. 231 275–327.
  • [10] Bismut, J.-M. (1977). Sur un problème de Dynkin. Z. Wahrsch. Verw. Gebiete 39 31–53.
  • [11] Bismut, J.-M. (1979). Contrôle de processus alternants et applications. Z. Wahrsch. Verw. Gebiete 47 241–288.
  • [12] Boetius, F. (2005). Bounded variation singular stochastic control and Dynkin game. SIAM J. Control Optim. 44 1289–1321 (electronic).
  • [13] Buckdahn, R. and Li, J. (2009). Probabilistic interpretation for systems of Isaacs equations with two reflecting barriers. NoDEA Nonlinear Differential Equations Appl. 16 381–420.
  • [14] Cattiaux, P. and Lepeltier, J.-P. (1990). Existence of a quasi-Markov Nash equilibrium for nonzero sum Markov stopping games. Stoch. Stoch. Rep. 30 85–103.
  • [15] Cosso, A. (2013). Stochastic differential games involving impulse controls and double-obstacle quasi-variational inequalities. SIAM J. Control Optim. 51 2102–2131.
  • [16] Cvitanić, J. and Karatzas, I. (1996). Backward stochastic differential equations with reflection and Dynkin games. Ann. Probab. 24 2024–2056.
  • [17] Dolinsky, Y. (2014). Hedging of game options under model uncertainty in discrete time. Electron. Commun. Probab. 19 no. 19, 11.
  • [18] Dynkin, E. (1967). Game variant of a problem on optimal stopping. Sov. Math., Dokl. 10 270–274.
  • [19] Ekren, I., Touzi, N. and Zhang, J. (2014). Optimal stopping under nonlinear expectation. Stochastic Process. Appl. 124 3277–3311.
  • [20] Ekren, I. and Zhang, J. (2016). Pseudo Markovian viscosity solutions of fully nonlinear degenerate PPDEs. Available at
  • [21] Ekström, E. (2006). Properties of game options. Math. Methods Oper. Res. 63 221–238.
  • [22] El Asri, B., Hamadène, S. and Wang, H. (2011). $L^{p}$-solutions for doubly reflected backward stochastic differential equations. Stoch. Anal. Appl. 29 907–932.
  • [23] Friedman, A. (1973). Stochastic games and variational inequalities. Arch. Ration. Mech. Anal. 51 321–346.
  • [24] Fukushima, M. and Taksar, M. (2002). Dynkin games via Dirichlet forms and singular control of one-dimensional diffusions. SIAM J. Control Optim. 41 682–699.
  • [25] Hamadène, S. (2006). Mixed zero-sum stochastic differential game and American game options. SIAM J. Control Optim. 45 496–518.
  • [26] Hamadène, S. and Hassani, M. (2005). BSDEs with two reflecting barriers: The general result. Probab. Theory Related Fields 132 237–264.
  • [27] Hamadène, S. and Hassani, M. (2014). The multi-player nonzero-sum Dynkin game in discrete time. Math. Methods Oper. Res. 79 179–194.
  • [28] Hamadène, S. and Hdhiri, I. (2006). Backward stochastic differential equations with two distinct reflecting barriers and quadratic growth generator. J. Appl. Math. Stoch. Anal. Art. ID 95818, 28.
  • [29] Hamadène, S. and Lepeltier, J.-P. (2000). Reflected BSDEs and mixed game problem. Stochastic Process. Appl. 85 177–188.
  • [30] Hamadène, S., Lepeltier, J.-P. and Wu, Z. (1999). Infinite horizon reflected backward stochastic differential equations and applications in mixed control and game problems. Probab. Math. Statist. 19 211–234.
  • [31] Hamadène, S. and Mohammed, H. (2014). The multiplayer nonzero-sum Dynkin game in continuous time. SIAM J. Control Optim. 52 821–835.
  • [32] Hamadène, S., Rotenstein, E. and Zălinescu, A. (2009). A generalized mixed zero-sum stochastic differential game and double barrier reflected BSDEs with quadratic growth coefficient. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 55 419–444.
  • [33] Hamadène, S. and Zhang, J. (2009/10). The continuous time nonzero-sum Dynkin game problem and application in game options. SIAM J. Control Optim. 48 3659–3669.
  • [34] Kallsen, J. and Kühn, C. (2004). Pricing derivatives of American and game type in incomplete markets. Finance Stoch. 8 261–284.
  • [35] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [36] Karatzas, I. and Wang, H. (2001). Connections between bounded variation control and Dynkin games. In Optimal Control and Partial Differential Equations (Volume in honor of A. Bensoussan) (Menaldi, J. L., Rofman, E. and Sulem, A., eds.) 363–373. IOS Press, Amsterdam.
  • [37] Karatzas, I. and Zamfirescu, I. M. (2008). Martingale approach to stochastic differential games of control and stopping. Ann. Probab. 36 1495–1527.
  • [38] Kifer, Y. (2000). Game options. Finance Stoch. 4 443–463.
  • [39] Kifer, Y. (2013). Dynkin games and Israeli options. ISRN Probability and Statistics 2013 Article ID 856458, 17 pages.
  • [40] Kobylanski, M., Quenez, M.-C. and de Campagnolle, M. R. (2014). Dynkin games in a general framework. Stochastics 86 304–329.
  • [41] Laraki, R. and Solan, E. (2005). The value of zero-sum stopping games in continuous time. SIAM J. Control Optim. 43 1913–1922 (electronic).
  • [42] Lepeltier, J.-P. and Maingueneau, M. A. (1984). Le jeu de Dynkin en théorie générale sans l’hypothèse de Mokobodski. Stochastics 13 25–44.
  • [43] Ma, J. and Cvitanić, J. (2001). Reflected forward-backward SDEs and obstacle problems with boundary conditions. J. Appl. Math. Stoch. Anal. 14 113–138.
  • [44] 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.
  • [45] Morimoto, H. (1984). Dynkin games and martingale methods. Stochastics 13 213–228.
  • [46] Nagai, H. (1987). Non-zero-sum stopping games of symmetric Markov processes. Probab. Theory Related Fields 75 487–497.
  • [47] Neveu, J. (1975). Discrete-Parameter Martingales, Revised ed. North-Holland Mathematical Library 10. North-Holland, Amsterdam.
  • [48] Nutz, M. (2012). A quasi-sure approach to the control of non-Markovian stochastic differential equations. Electron. J. Probab. 17 1–23.
  • [49] Nutz, M. and Zhang, J. (2015). Optimal stopping under adverse nonlinear expectation and related games. Ann. Appl. Probab. 25 2503–2534.
  • [50] Ohtsubo, Y. (1987). A nonzero-sum extension of Dynkin’s stopping problem. Math. Oper. Res. 12 277–296.
  • [51] Peng, S. (2007). G-Brownian motion and dynamic risk measure under volatility uncertainty. Available at
  • [52] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 2: Itô Calculus. Cambridge Univ. Press, Cambridge. Reprint of the second (1994) edition.
  • [53] Rosenberg, D., Solan, E. and Vieille, N. (2001). Stopping games with randomized strategies. Probab. Theory Related Fields 119 433–451.
  • [54] Snell, J. L. (1952). Applications of martingale system theorems. Trans. Amer. Math. Soc. 73 293–312.
  • [55] Soner, H., Touzi, N. and Zhang, J. (2013). Dual formulation of second order target problems. Ann. Appl. Probab. 23 308–347.
  • [56] Stettner, Ł. (1982/83). Zero-sum Markov games with stopping and impulsive strategies. Appl. Math. Optim. 9 1–24.
  • [57] Stroock, D. W. and Varadhan, S. R. S. (2006). Multidimensional Diffusion Processes. Springer, Berlin. Reprint of the 1997 edition.
  • [58] Taksar, M. I. (1985). Average optimal singular control and a related stopping problem. Math. Oper. Res. 10 63–81.
  • [59] Touzi, N. and Vieille, N. (2002). Continuous-time Dynkin games with mixed strategies. SIAM J. Control Optim. 41 1073–1088.
  • [60] Xu, M. (2007). Reflected backward SDEs with two barriers under monotonicity and general increasing conditions. J. Theoret. Probab. 20 1005–1039.
  • [61] Yasuda, M. (1985). On a randomized strategy in Neveu’s stopping problem. Stochastic Process. Appl. 21 159–166.
  • [62] Yin, J. (2012). Reflected backward stochastic differential equations with two barriers and Dynkin games under Knightian uncertainty. Bull. Sci. Math. 136 709–729.