On the Controller-Stopper Problems with Controlled Jumps

  • Erhan Bayraktar
  • Jiaqi Li


We analyze the continuous time zero-sum and cooperative controller-stopper games of Karatzas and Sudderth (Ann Probab 29(3):1111–1127, 2001), Karatzas and Zamfirescu (Ann Probab 36(4):1495–1527, 2008) and Karatzas and Zamfirescu (Appl Math Optim 53(2):163–184, 2006) when the volatility of the state process is controlled as in Bayraktar and Huang (SIAM J Control Optim 51(2):1263–1297, 2013) but additionally when the state process has controlled jumps. We perform this analysis by first resolving the stochastic target problems [of Soner and Touzi (SIAM J Control Optim 41(2):404–424, 2002; J Eur Math Soc 4(3):201–236, 2002)] with a cooperative or a non-cooperative stopper and then embedding the original problem into the latter set-up. Unlike in Bayraktar and Huang (SIAM J Control Optim 51(2):1263–1297, 2013) our analysis relies crucially on the Stochastic Perron method of Bayraktar and Sîrbu (SIAM J Control Optim 51(6):4274–4294, 2013) but not the dynamic programming principle, which is difficult to prove directly for games.


Controller-stopper problems Stochastic target problems Stochastic Perron’s method Viscosity solution 

Mathematics Subject Classification

Primary 93E20 Secondary 49L25 60J75 60G40 



E. Bayraktar is supported in part by the National Science Foundation under Grant DMS-1613170 and the Susan M. Smith Professorship.


  1. 1.
    Barles, G., Imbert, C.: Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(3), 567–585 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bayraktar, E., Huang, Y.-J.: On the multidimensional controller-and-stopper games. SIAM J. Control Optim. 51(2), 1263–1297 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bayraktar, E., Li, J.: Stochastic perron for stochastic target problems. J. Optim. Theory Appl. 170(3), 1026–1054 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bayraktar, E., Li, J.: Stochastic Perron for stochastic target games. Ann. Appl. Probab. 26(2), 1082–1110 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bayraktar, E., Sîrbu, M.: Stochastic Perron’s method and verification without smoothness using viscosity comparison: the linear case. Proc. Am. Math. Soc. 140(10), 3645–3654 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bayraktar, E., Sîrbu, M.: Stochastic Perron’s method for Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 51(6), 4274–4294 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bayraktar, E., Yao, S.: Optimal stopping for non-linear expectations-Part II. Stoch. Process. Appl. 121(2), 212–264 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bayraktar, E., Yao, S.: On the robust optimal stopping problem. SIAM J. Control Optim. 52(5), 3135–3175 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bayraktar, E., Yao, S.: On the robust Dynkin game. Ann. Appl. Probab. 27(3), 1702–1755 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bayraktar, E., Zhang, Y.: Stochastic Perron’s method for the probability of lifetime ruin problem under transaction costs. SIAM J. Control Optim. 53(1), 91–113 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bayraktar, E., Karatzas, I., Yao, S.: Optimal stopping for dynamic convex risk measures. Illinois J. Math. 54(3), 1025–1067 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Bayraktar, E., Cosso, A., Pham, H.: Robust feedback switching control: dynamic programming and viscosity solutions. SIAM J. Control Optim. 54(5), 2594–2628 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Belomestny, D., Krätschmer, V.: Optimal stopping under model uncertainty: randomized stopping times approach. Ann. Appl. Probab. 26(2), 1260–1295 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bensoussan, A., Lions, J.-L.: Applications of Variational Inequalities in Stochastic Control. Studies in Mathematics and its Applications, vol. 12. North-Holland Publishing Co., Amsterdam (1982). Translated from the FrenchGoogle Scholar
  17. 17.
    Bouchard, B.: Stochastic targets with mixed diffusion processes and viscosity solutions. Stoch. Process. Appl. 101(2), 273–302 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bouchard, B., Dang, N.M.: Optimal control versus stochastic target problems: an equivalence result. Syst. Control Lett. 61(2), 343–346 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Bouchard, B., Nutz, M.: Stochastic target games and dynamic programming via regularized viscosity solutions. Math. Oper. Res. 41(1), 109–124 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Bouchard, B., Vu, T.N.: The obstacle version of the geometric dynamic programming principle: application to the pricing of American options under constraints. Appl. Math. Optim. 61(2), 235–265 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Bouchard, B., Elie, R., Touzi, N.: Stochastic target problems with controlled loss. SIAM J. Control Optim. 48(5), 3123–3150 (2009/2010)Google Scholar
  22. 22.
    Bouchard, B., Moreau, L., Nutz, M.: Stochastic target games with controlled loss. Ann. Appl. Probab. 24(3), 899–934 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ceci, C., Bassan, B.: Mixed optimal stopping and stochastic control problems with semicontinuous final reward for diffusion processes. Stoch. Stoch. Rep. 76(4), 323–337 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Cheng, X., Riedel, F.: Optimal stopping under ambiguity in continuous time. Math. Financ. Econ. 7(1), 29–68 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Cohen, S.N., Elliott, R.J.: Stochastic Calculus and Applications, 2nd edn. Probability and its Applications. Springer, Cham (2015)CrossRefzbMATHGoogle Scholar
  26. 26.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Dmitry, B.: Stochastic Perron’s method for optimal control problems with state constraints. Electron. Commun. Probab. 19(73), 15 (2014)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Ekren, I., Touzi, N., Zhang, J.: Optimal stopping under nonlinear expectation. Stoch. Process. Appl. 124(10), 3277–3311 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    El Karoui, N.: Les aspects probabilistes du contrôle stochastique. Ninth Saint Flour Probability Summer School–1979 (Saint Flour. Lecture Notes in Mathematics, vol. 876. Springer, Berlin 1981, 73–238 (1979)Google Scholar
  30. 30.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Graduate Texts in Mathematics. Springer, Berlin (1991)zbMATHGoogle Scholar
  32. 32.
    Karatzas, I., Sudderth, W.D.: The controller-and-stopper game for a linear diffusion. Ann. Probab. 29(3), 1111–1127 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Karatzas, I., Zamfirescu, I.-M.: Martingale approach to stochastic control with discretionary stopping. Appl. Math. Optim. 53(2), 163–184 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Karatzas, I., Zamfirescu, I.-M.: Martingale approach to stochastic differential games of control and stopping. Ann. Probab. 36(4), 1495–1527 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Krätschmer, V., Schoenmakers, J.: Representations for optimal stopping under dynamic monetary utility functionals. SIAM J. Financ. Math. 1(1), 811–832 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Krylov, NV.: Controlled Diffusion Processes. Applications of Mathematics, vol. 14, Springer, New York (1980). Translated from the Russian by A. B. AriesGoogle Scholar
  37. 37.
    Maitra, A.P., Sudderth, W.D.: Discrete Gambling and Stochastic Games. Applications of Mathematics, vol. 32. Springer, New York (1996)Google Scholar
  38. 38.
    Maitra, A.P., Sudderth, W.D.: The Gambler and the Stopper, Statistics, Probability and Game Theory. IMS Lecture Notes Monograph Series, vol. 30. Institute of Mathematical Statistics, Hayward, CA, pp. 191–208 (1996)Google Scholar
  39. 39.
    Moreau, L.: Stochastic target problems with controlled loss in jump diffusion models. SIAM J. Control Optim. 49(6), 2577–2607 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Nutz, M., Zhang, J.: Optimal stopping under adverse nonlinear expectation and related games. Ann. Appl. Probab. 25(5), 2503–2534 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Pham, H.: Optimal stopping of controlled jump diffusion processes: a viscosity solution approach. J. Math. Syst. Estim. Control 8(1), 27 (1998)MathSciNetGoogle Scholar
  42. 42.
    Riedel, F.: Optimal stopping with multiple priors. Econometrica 77(3), 857–908 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Rokhlin, D.B.: Verification by stochastic Perron’s method in stochastic exit time control problems. J. Math. Anal. Appl. 419(1), 433–446 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Rokhlin, D.B., Mironenko, G.: Regular finite fuel stochastic control problems with exit time. Math. Methods Oper. Res. 84(1), 105–127 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Sîrbu, M.: Stochastic Perron’s method and elementary strategies for zero-sum differential games. SIAM J. Control Optim. 52(3), 1693–1711 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Soner, H.M., Touzi, N.: Dynamic programming for stochastic target problems and geometric flows. J. Eur. Math. Soc. 4(3), 201–236 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Soner, H.M., Touzi, N.: Stochastic target problems, dynamic programming, and viscosity solutions. SIAM J. Control Optim. 41(2), 404–424 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Touzi, N.: Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE. Fields Institute Monographs, vol. 29. Fields Institute for Research in Mathematical Sciences, Toronto, ON. Springer, New York (2013). With Chapter 13 by Angès TourinGoogle Scholar
  49. 49.
    Veraguas, J.B., Tangpi, L.: On the dynamic representation of some time-inconsistent risk measures in a Brownian filtration. ArXiv e-prints (2016)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.University of MichiganAnn ArborUSA
  2. 2.Goldman SachsNew YorkUSA

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