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Bayraktar, Erhan; Li, Jiaqi

Stochastic Perron for stochastic target games. (English) Zbl 1337.93100

Ann. Appl. Probab. 26, No. 2, 1082-1110 (2016).
Summary: We extend the stochastic Perron method to analyze the framework of stochastic target games, in which one player tries to find a strategy such that the state process almost surely reaches a given target no matter which action is chosen by the other player. Within this framework, our method produces a viscosity sub-solution (super-solution) of a Hamilton-Jacobi-Bellman (HJB) equation. We then characterize the value function as a viscosity solution to the HJB equation using a comparison result and a byproduct to obtain the dynamic programming principle.

Cited in 5 Documents

MSC:

93E20 Optimal stochastic control
49L20 Dynamic programming in optimal control and differential games
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
60G46 Martingales and classical analysis
60H30 Applications of stochastic analysis (to PDEs, etc.)
91A25 Dynamic games

Keywords:

stochastic target problem; stochastic Perron method; viscosity solutions; geometric dynamic programming principle
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\textit{E. Bayraktar} and \textit{J. Li}, Ann. Appl. Probab. 26, No. 2, 1082--1110 (2016; Zbl 1337.93100)
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References:

[1] Bayraktar, E., Cosso, A. and Pham, H. (2014). Robust feedback switching control: Dynamic programming and viscosity solutions. Available at . arXiv:1409.6233 · Zbl 1347.49042
[2] Bayraktar, E. and Huang, Y.-J. (2013). On the multidimensional controller-and-stopper games. SIAM J. Control Optim. 51 1263-1297. · Zbl 1268.49045
[3] Bayraktar, E. and Sîrbu, M. (2012). Stochastic Perron’s method and verification without smoothness using viscosity comparison: The linear case. Proc. Amer. Math. Soc. 140 3645-3654. · Zbl 1279.60056
[4] Bayraktar, E. and Sîrbu, M. (2013). Stochastic Perron’s method for Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 51 4274-4294. · Zbl 1285.49019
[5] 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. · Zbl 1321.60080
[6] Bayraktar, E. and Zhang, Y. (2015). Stochastic Perron’s method for the probability of lifetime ruin problem under transaction costs. SIAM J. Control Optim. 53 91-113. · Zbl 1343.93094
[7] Bouchard, B., Moreau, L. and Nutz, M. (2014). Stochastic target games with controlled loss. Ann. Appl. Probab. 24 899-934. · Zbl 1290.49075
[8] Bouchard, B. and Nutz, M. (2015). Stochastic target games and dynamic programming via regularized viscosity solutions. Math. Oper. Res. To appear. Available at . arXiv:1307.5606 · Zbl 1334.93178
[9] 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. · Zbl 0755.35015
[10] Pham, H. (2009). Continuous-Time Stochastic Control and Optimization with Financial Applications. Stochastic Modelling and Applied Probability 61 . Springer, Berlin. · Zbl 1165.93039
[11] Rokhlin, D. B. (2014). Stochastic Perron’s method for optimal control problems with state constraints. Electron. Commun. Probab. 19 1-15. · Zbl 1310.93085
[12] Rokhlin, D. B. (2014). Verification by stochastic Perron’s method in stochastic exit time control problems. J. Math. Anal. Appl. 419 433-446. · Zbl 1297.35209
[13] Sîrbu, M. (2014). A note on the strong formulation of stochastic control problems with model uncertainty. Electron. Commun. Probab. 19 1-10. · Zbl 1325.49025
[14] Sîrbu, M. (2014). Stochastic Perron’s method and elementary strategies for zero-sum differential games. SIAM J. Control Optim. 52 1693-1711. · Zbl 1409.91029
[15] Soner, H. M. and Touzi, N. (2002). Dynamic programming for stochastic target problems and geometric flows. J. Eur. Math. Soc. ( JEMS ) 4 201-236. · Zbl 1003.49003
[16] Soner, H. M. and Touzi, N. (2002). Stochastic target problems, dynamic programming, and viscosity solutions. SIAM J. Control Optim. 41 404-424. · Zbl 1011.49019
[17] Touzi, N. (2013). Optimal Stochastic Control , Stochastic Target Problems , and Backward SDE. Fields Institute Monographs 29 . Springer, New York. · Zbl 1256.93008
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