A Zero-Sum Stochastic Differential Game with Impulses, Precommitment, and Unrestricted Cost Functions

  • Parsiad Azimzadeh


We study a zero-sum stochastic differential game (SDG) in which one controller plays an impulse control while their opponent plays a stochastic control. We consider an asymmetric setting in which the impulse player commits to, at the start of the game, performing less than q impulses (q can be chosen arbitrarily large). In order to obtain the uniform continuity of the value functions, previous works involving SDGs with impulses assume the cost of an impulse to be decreasing in time. Our work avoids such restrictions by requiring impulses to occur at rational times. We establish that the resulting game admits a value, and in turn, the existence and uniqueness of viscosity solutions to an associated Hamilton-Jacobi-Bellman-Isaacs quasi-variational inequality.


Zero-sum stochastic differential game Impulse control Quasi-variational inequalities Viscosity solutions 

Mathematics Subject Classification

49L20 49L25 91A23 91A15 



The author wishes to express his deepest gratitude to Catherine Rainer (Université de Brest) for all of her help, support, and for pointing out the use of delay strategies to establish the pointwise inequality \(v^{-}\leqslant v^{+}\). The author also thanks Andrea Cosso (Politecnico di Milano) for discussion regarding DPPs, Christine Grün (Université Toulouse 1 Capitole) for discussion on games, and Nizar Touzi (École Polytechnique) for discussion regarding comparison.


  1. 1.
    Aïd, R., Basei, M., Callegaro, G., Campi, L., Vargiolu, T.: Nonzero-sum stochastic differential games with impulse controls: a verification theorem with applications. arXiv preprint arXiv:1605.00039 (2016)
  2. 2.
    Aubin, J.-P., Seube, N.: Conditional viability for impulse differential games. Ann. Oper. Res. 137, 269–297 (2005). doi: 10.1007/s10479-005-2261-8. ISSN 0254-5330. Contributions to the theory of games
  3. 3.
    Azimzadeh, P., Forsyth, P.A.: Weakly chained matrices, policy iteration, and impulse control. SIAM J. Numer. Anal. 54(3), 1341–1364 (2016). doi: 10.1137/15M1043431. ISSN 0036-1429
  4. 4.
    Bayraktar, E., Huang, Y.-J.: On the multidimensional controller-and-stopper games. SIAM J. Control Optim. 51(2), 1263–1297 (2013). doi: 10.1137/110847329
  5. 5.
    Bayraktar, E., Sîrbu, M.: Stochastic Perron’s method and verification without smoothness using viscosity comparison: obstacle problems and Dynkin games. Proc. Am. Math. Soc. 142(4), 1399–1412 (2014). doi: 10.1090/S0002-9939-2014-11860-0. ISSN 0002-9939
  6. 6.
    Bayraktar, E., Emmerling, T., Menaldi, J.-L.: On the impulse control of jump diffusions. SIAM J. Control Optim. 51(3), 2612–2637 (2013). doi: 10.1137/120863836. ISSN 0363-0129
  7. 7.
    Bayraktar, E., Cosso, A., Pham, H.: Robust feedback switching control: dynamic programming and viscosity solutions. SIAM J. Control Optim. 54(5), 2594–2628 (2016). doi: 10.1137/15M1046903. ISSN 0363-0129
  8. 8.
    Belak, C., Christensen, S.: Utility maximization in a factor model with constant and proportional costs (2016)Google Scholar
  9. 9.
    Belak, C., Christensen, S., Seifried, F.T.: A general verification result for stochastic impulse control problems. SIAM J. Control Optim. 55(2), 627–649 (2017). doi: 10.1137/16M1082822. ISSN 0363-0129
  10. 10.
    Berestycki, H., Monneau, R., Scheinkman, J.: A non-local free boundary problem arising in a theory of financial bubbles. Philos. Trans. R. Soc. Lond. A 372(2028), 20130404, 36 (2014). doi: 10.1098/rsta.2013.0404. ISSN 1364-503X
  11. 11.
    Bernhard, P.: On the singularities of an impulsive differential game arising in mathematical finance. Int. Game Theory Rev. 8(2), 219–229 (2006). doi: 10.1142/S0219198906000874. ISSN 0219-1989
  12. 12.
    Bouchard, B., Touzi, N.: Weak dynamic programming principle for viscosity solutions. SIAM J. Control Optim. 49(3), 948–962 (2011). doi: 10.1137/090752328. ISSN 0363-0129
  13. 13.
    Bruder, B., Pham, H.: Impulse control problem on finite horizon with execution delay. Stoch. Process. Appl. 119(5), 1436–1469 (2009). doi: 10.1016/ ISSN 0304-4149
  14. 14.
    Buckdahn, R., Li, J.: Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations. SIAM J. Control Optim. 47(1), 444–475 (2008). doi: 10.1137/060671954. ISSN 0363-0129
  15. 15.
    Buckdahn, R., Cardaliaguet, P., Rainer, C.: Nash equilibrium payoffs for nonzero-sum stochastic differential games. SIAM J. Control Optim. 43(2), 624–642 (electronic) (2004). doi: 10.1137/S0363012902411556. ISSN 0363-0129
  16. 16.
    Cardaliaguet, P., Rainer, C.: Stochastic differential games with asymmetric information. Appl. Math. Optim. 59(1), 1–36 (2009). doi: 10.1007/s00245-008-9042-0. ISSN 0095-4616
  17. 17.
    Chen, Y.-S.A., Guo, X.: Impulse control of multidimensional jump diffusions in finite time horizon. SIAM J. Control Optim. 51(3), 2638–2663 (2013). doi: 10.1137/110854205. ISSN 0363-0129
  18. 18.
    Chevalier, E., Ly Vath, V., Scotti, S., Roch, A.: Optimal execution cost for liquidation through a limit order market. Int. J. Theor. Appl. Finance 19(1), 1650004, 26 (2016). doi: 10.1142/S0219024916500047. ISSN 0219-0249
  19. 19.
    Chikrii, A.A., Matychyn, I.I., Chikrii, K.A.: Differential games with impulse control. In: Advances in Dynamic Game Theory. Annals of the International Society of Dynamic Games, vol. 9, pp. 37–55. Birkhäuser, Boston (2007). doi: 10.1007/978-0-8176-4553-3_2
  20. 20.
    Cosso, A.: Stochastic differential games involving impulse controls and double-obstacle quasi-variational inequalities. SIAM J. Control Optim. 51(3), 2102–2131 (2013). doi: 10.1137/120880094. ISSN 0363-0129
  21. 21.
    Crandall, M.G., Ishii, H.: The maximum principle for semicontinuous functions. Differ. Integr. Equ. 3(6), 1001–1014 (1990). ISSN 0893-4983Google Scholar
  22. 22.
    Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992). doi: 10.1090/S0273-0979-1992-00266-5. ISSN 0273-0979
  23. 23.
    Dumitrescu, R., Quenez, M.-C., Sulem, A.: Mixed generalized Dynkin game and stochastic control in a Markovian framework. Stochastics 89(1), 400–429 (2017). doi: 10.1080/17442508.2016.1230614. ISSN 1744-2508
  24. 24.
    Elliott, R.J., Kalton, N.J.: The Existence of Value in Differential Games. American Mathematical Society, Providence (1972). Memoirs of the American Mathematical Society, No. 126Google Scholar
  25. 25.
    Evans, L.C., Souganidis, P.E.: Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations. Indiana Univ. Math. J. 33(5), 773–797 (1984). doi: 10.1512/iumj.1984.33.33040. ISSN 0022-2518
  26. 26.
    Fleming, W.H., Souganidis, P.E.: On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38(2), 293–314 (1989). doi: 10.1512/iumj.1989.38.38015. ISSN 0022-2518
  27. 27.
    Hamadène, S., Lepeltier, J.-P.: Zero-sum stochastic differential games and backward equations. Syst. Control Lett. 24(4), 259–263 (1995). doi: 10.1016/0167-6911(94)00011-J. ISSN 0167-6911
  28. 28.
    Hamadène, S., Wang, H.: The mixed zero-sum stochastic differential game in the model with jumps. In: Advances in Dynamic Games. Annals of the International Society of Dynamic Games, vol. 11, pp. 83–110. Birkhäuser/Springer, New York (2011). doi: 10.1007/978-0-8176-8089-3_5
  29. 29.
    Hamadène, S., Lepeltier, J.-P., Peng, S.: BSDEs with continuous coefficients and stochastic differential games. Backward Stochastic Differential Equations (Paris, 1995–1996). Pitman Research Notes in Mathematics Series, vol. 364, pp. 115–128. Longman, Harlow (1997)Google Scholar
  30. 30.
    Isaacs, R.: Differential Games. Control and Optimization. A Mathematical Theory with Applications to Warfare and Pursuit. Wiley, New York (1965)zbMATHGoogle Scholar
  31. 31.
    Ishii, K.: Viscosity solutions of nonlinear second order elliptic PDEs associated with impulse control problems. Funkcial. Ekvac. 36(1), 123–141 (1993). ISSN 0532-8721.
  32. 32.
    Kaczor, W.J., Nowak, M.T.: Problems in Mathematical Analysis. I: Real Numbers, Sequences and Series. Student Mathematical Library, vol. 4. American Mathematical Society, Providence (2000). doi: 10.1090/stml/004. ISBN 0-8218-2050-8
  33. 33.
    Kharroubi, I., Huyên Pham, J.M., Zhang, J.: Backward SDEs with constrained jumps and quasi-variational inequalities. Ann. Probab. 38(2), 794–840 (2010). doi: 10.1214/09-AOP496. ISSN 0091-1798
  34. 34.
    Ly Vath, V., Mnif, M., Pham, H.: A model of optimal portfolio selection under liquidity risk and price impact. Finance Stoch. 11(1), 51–90 (2007). doi: 10.1007/s00780-006-0025-1. ISSN 0949-2984
  35. 35.
    Nutz, M., Zhang, J.: Optimal stopping under adverse nonlinear expectation and related games. Ann. Appl. Probab. 25(5), 2503–2534 (2015). doi: 10.1214/14-AAP1054. ISSN 1050-5164
  36. 36.
    Øksendal, B.: Stochastic differential Equations. Universitext, 6th edn. Springer, Berlin (2003). doi: 10.1007/978-3-642-14394-6. ISBN 3-540-04758-1. An introduction with applications
  37. 37.
    Pham, T., Zhang, J.: Two person zero-sum game in weak formulation and path dependent Bellman-Isaacs equation. SIAM J. Control Optim. 52(4), 2090–2121 (2014). doi: 10.1137/120894907. ISSN 0363-0129
  38. 38.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, 2nd edn. Springer, Berlin (1994). ISBN 3-540-57622-3Google Scholar
  39. 39.
    Seydel, R.C.: Existence and uniqueness of viscosity solutions for QVI associated with impulse control of jump-diffusions. Stochastic Process. Appl. 119(10), 3719–3748 (2009). doi: 10.1016/ ISSN 0304-4149
  40. 40.
    Shaiju, A.J., Dharmatti, S.: Differential games with continuous, switching and impulse controls. Nonlinear Anal. 63(1), 23–41 (2005). doi: 10.1016/ ISSN 0362-546X
  41. 41.
    Sîrbu, M.: Stochastic Perron’s method and elementary strategies for zero-sum differential games. SIAM J. Control Optim. 52(3), 1693–1711 (2014). doi: 10.1137/130929965. ISSN 0363-0129
  42. 42.
    Tang, S.J., Yong, J.M.: Finite horizon stochastic optimal switching and impulse controls with a viscosity solution approach. Stocha. Stoch. Rep. 45(3–4), 145–176 (1993). doi: 10.1080/17442509308833860. ISSN 1045-1129
  43. 43.
    Tang, S., Hou, S.-H.: Switching games of stochastic differential systems. SIAM J. Control Optim. 46(3), 900–929 (2007). doi: 10.1137/050642204. ISSN 0363-0129
  44. 44.
    Touzi, N.: Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE. Fields Institute Monographs, vol. 29. Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto (2013). ISBN 978-1-4614-4285-1; 978-1-4614-4286-8. doi: 10.1007/978-1-4614-4286-8 (Chapter 13 by Angès Tourin)
  45. 45.
    Yong, J.M.: Zero-sum differential games involving impulse controls. Appl. Math. Optim. 29(3), 243–261 (1994). doi: 10.1007/BF01189477. ISSN 0095-4616
  46. 46.
    Zhang, F.: Stochastic differential games involving impulse controls. ESAIM Control Optim. Calc. Var. 17(3), 749–760 (2011). doi: 10.1051/cocv/2010023. ISSN 1292-8119

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations