The Annals of Probability

Value in mixed strategies for zero-sum stochastic differential games without Isaacs condition

Rainer Buckdahn, Juan Li, and Marc Quincampoix

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In the present work, we consider 2-person zero-sum stochastic differential games with a nonlinear pay-off functional which is defined through a backward stochastic differential equation. Our main objective is to study for such a game the problem of the existence of a value without Isaacs condition. Not surprising, this requires a suitable concept of mixed strategies which, to the authors’ best knowledge, was not known in the context of stochastic differential games. For this, we consider nonanticipative strategies with a delay defined through a partition $\pi$ of the time interval $[0,T]$. The underlying stochastic controls for the both players are randomized along $\pi$ by a hazard which is independent of the governing Brownian motion, and knowing the information available at the left time point $t_{j-1}$ of the subintervals generated by $\pi$, the controls of Players 1 and 2 are conditionally independent over $[t_{j-1},t_{j})$. It is shown that the associated lower and upper value functions $W^{\pi}$ and $U^{\pi}$ converge uniformly on compacts to a function $V$, the so-called value in mixed strategies, as the mesh of $\pi$ tends to zero. This function $V$ is characterized as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman–Isaacs equation.

Article information

Ann. Probab., Volume 42, Number 4 (2014), 1724-1768.

First available in Project Euclid: 3 July 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 49N70: Differential games 49L25: Viscosity solutions
Secondary: 91A23: Differential games [See also 49N70] 60H10: Stochastic ordinary differential equations [See also 34F05]

2-person zero-sum stochastic differential game Isaacs condition viscosity solution value function backward stochastic differential equations dynamic programming principle randomized controls


Buckdahn, Rainer; Li, Juan; Quincampoix, Marc. Value in mixed strategies for zero-sum stochastic differential games without Isaacs condition. Ann. Probab. 42 (2014), no. 4, 1724--1768. doi:10.1214/13-AOP849.

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  • [1] Buckdahn, R., Cardaliaguet, P. and Quincampoix, M. (2011). Some recent aspects of differential game theory. Dyn. Games Appl. 1 74–114.
  • [2] Buckdahn, R., Cardaliaguet, P. and Rainer, C. (2004). Nash equilibrium payoffs for nonzero-sum stochastic differential games. SIAM J. Control Optim. 43 624–642 (electronic).
  • [3] Buckdahn, R. and Li, J. (2008). Stochastic differential games and viscosity solutions of Hamilton–Jacobi–Bellman–Isaacs equations. SIAM J. Control Optim. 47 444–475.
  • [4] Buckdahn, R., Li, J. and Quincampoix, M. (2013). Value function of differential games without Isaacs conditions. An approach with nonanticipative mixed strategies. Internat. J. Game Theory 42 989–1020.
  • [5] Carbone, R., Ferrario, B. and Santacroce, M. (2008). Backward stochastic differential equations driven by càdlàg martingales. Theory Probab. Appl. 52 304–314.
  • [6] Crandall, M. G., Ishii, H. and Lions, P.-L. (1992). User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 1–67.
  • [7] Dellacherie, C. (1977). Sur L’existence de Certains Essinf et Esssup de Familles de Processus Mesurables. Sem. Probab. XII, Lecture Notes in Math. 649. Springer, Berlin.
  • [8] Dunford, N. and Schwartz, J. T. (1957). Linear Operators. Part I: General Theory. Wiley, New York.
  • [9] Fleming, W. H. and Hernández-Hernández, D. (2011). On the value of stochastic differential games. Commun. Stoch. Anal. 5 341–351.
  • [10] 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.
  • [11] Hamadene, S., Lepeltier, J. P. and Peng, S. (1997). BSDEs with continuous coefficients and stochastic differential games. In Backward Stochastic Differential Equations (Paris, 19951996) (N. El Karoui and L. Mazliak, eds.). Pitman Res. Notes Math. Ser. 364 115–128. Longman, Harlow.
  • [12] Isaacs, R. (1965). Differential Games. A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization. Wiley, New York.
  • [13] Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Applications of Mathematics (New York) 39. Springer, New York.
  • [14] Krasovskiĭ, N. N. and Subbotin, A. I. (1988). Game-Theoretical Control Problems. Springer, New York.
  • [15] Krylov, N. V. (2012). On the dynamic programming principle for uniformly non-degenerate stochastic differential games in domains. Available at
  • [16] Krylov, N. V. (2012). On the dynamic programming principle for uniformly non-degenerate stochastic differential games in domains and the Isaacs equations. Available at
  • [17] Pardoux, É. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55–61.
  • [18] Peng, S. (1997). BSDE and Stochastic Optimizations; Topics in Stochastic Analysis (J. Yan, S. Peng, S. Fang and L. Wu, eds.). Science Press, Beijing.
  • [19] Strömberg, T. (2008). Exponentially growing solutions of parabolic Isaacs’ equations. J. Math. Anal. Appl. 348 337–345.
  • [20] Subbotin, A. I. and Chentsov, A. G. (1981). Optimizatsiya Garantii v Zadachakh Upravleniya. Nauka, Moscow.
  • [21] Święch, A. (1996). Another approach to the existence of value functions of stochastic differential games. J. Math. Anal. Appl. 204 884–897.