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Game value for a pursuit-evasion differential game problem in a Hilbert space
Zero-sum games for pure jump processes with risk-sensitive discounted cost criteria
| 1. | Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, Assam - 781039, India |
| 2. | Department of Mathematics, Indian Institute of Science Education and Research Pune, Pune, Maharashtra - 411008, India |
In this paper we study zero-sum stochastic games for pure jump processes on a general state space with risk sensitive discounted criteria. We establish a saddle point equilibrium in Markov strategies for bounded cost function. We achieve our results by studying relevant Hamilton-Jacobi-Isaacs equations.
References:
| [1] |
R. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957.
|
| [2] |
V. E. Beneš,
Existence of optimal strategies based on specified information, for a class of stochastic decision problems, SIAM J. Control, 8 (1970), 179-188.
doi: 10.1137/0308012. |
| [3] |
K. Fan,
Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 121-126.
doi: 10.1073/pnas.38.2.121. |
| [4] |
M. K. Ghosh, K. S. Kumar and C. Pal,
Zero-sum risk-sensitive stochastic games for continuous time Markov chains, Stoch. Anal. Appl., 34 (2016), 835-851.
doi: 10.1080/07362994.2016.1180995. |
| [5] |
M. K. Ghosh and S. Saha,
Risk-sensitive control of continuous time Markov chains, Stochastics, 86 (2014), 655-675.
doi: 10.1080/17442508.2013.872644. |
| [6] |
X. Guo,
Continuous-time Markov decision processes with discounted rewards: The case of Polish spaces, Math. Oper. Res., 32 (2007), 73-87.
doi: 10.1287/moor.1060.0210. |
| [7] |
X. Guo and O. Hernández-Lerma,
Nonzero-sum games for continuous-time Markov chains with unbounded discounted payoffs, J. Appl. Probab., 42 (2005), 303-320.
doi: 10.1239/jap/1118777172. |
| [8] |
X. Guo and O. Hernández-Lerma,
Zero-sum games for continuous-time jump Markov processes in Polish spaces: Discounted payoffs, Adv. in Appl. Probab., 39 (2007), 645-668.
doi: 10.1017/S0001867800001981. |
| [9] |
X. Guo and O. Hernández-Lerma,
Zero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates, J. Appl. Probab., 40 (2003), 327-345.
doi: 10.1017/S0021900200019331. |
| [10] |
X. Guo and Z.-W. Liao,
Risk-sensitive discounted continuous-time Markov decision processes with unbounded rates, SIAM J. Control Optim., 57 (2019), 3857-3883.
doi: 10.1137/18M1222016. |
| [11] |
X. Guo and Y. Zhang,
On risk-sensitive piecewise deterministic Markov decision processes, Appl. Math. Optim., 81 (2020), 685-710.
doi: 10.1007/s00245-018-9485-x. |
| [12] |
O. Hernández-Lerma and J. B. Lasserre, Further Topics on Discrete-Time Markov Control Processes, Applications of Mathematics (New York), 42, Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-0561-6. |
| [13] |
A. S. Nowak, Notes on risk-sensitive Nash equilibria, in Advances in Dynamic Games, Ann. Internat. Soc. Dynam. Games, 7, Birkhäuser Boston, Boston, MA, 2005, 95–109.
doi: 10.1007/0-8176-4429-6_5. |
| [14] |
C. Pal and S. Pradhan,
Risk sensitive control of pure jump processes on a general state space, Stochastics, 91 (2019), 155-174.
doi: 10.1080/17442508.2018.1521413. |
| [15] |
K. Suresh Kumar and C. Pal,
Risk-sensitive control of pure jump process on countable space with near monotone cost, Appl. Math. Optim., 68 (2013), 311-331.
doi: 10.1007/s00245-013-9208-2. |
| [16] |
K. Suresh Kumar and C. Pal,
Risk-sensitive ergodic control of continuous time Markov processes with denumerable state space, Stoch. Anal. Appl., 33 (2015), 863-881.
doi: 10.1080/07362994.2015.1050674. |
| [17] |
Q. Wei,
Zero-sum games for continuous-time Markov jump processes with risk-sensitive finite-horizon cost criterion, Oper. Res. Lett., 46 (2018), 69-75.
doi: 10.1016/j.orl.2017.11.008. |
| [18] |
Q. Wei and X. Chen,
Stochastic games for continuous-time jump processes under finite-horizon payoff criterion, Appl. Math. Optim., 74 (2016), 273-301.
doi: 10.1007/s00245-015-9314-4. |
| [19] |
P. Whittle, Risk-Sensitive Optimal Control, Wiley-Interscience Series in Systems and Optimization, John Wiley & Sons, Ltd., Chichester, 1990. |
| [20] |
W. Zhang,
Continuous-time constrained stochastic games under the discounted cost criteria, Appl. Math. Optim., 77 (2018), 275-296.
doi: 10.1007/s00245-016-9374-0. |
| [21] |
Y. Zhang,
Continuous-time Markov decision processes with exponential utility, SIAM J. Control Optim., 55 (2017), 2636-2660.
doi: 10.1137/16M1086261. |
show all references
References:
| [1] |
R. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957.
|
| [2] |
V. E. Beneš,
Existence of optimal strategies based on specified information, for a class of stochastic decision problems, SIAM J. Control, 8 (1970), 179-188.
doi: 10.1137/0308012. |
| [3] |
K. Fan,
Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 121-126.
doi: 10.1073/pnas.38.2.121. |
| [4] |
M. K. Ghosh, K. S. Kumar and C. Pal,
Zero-sum risk-sensitive stochastic games for continuous time Markov chains, Stoch. Anal. Appl., 34 (2016), 835-851.
doi: 10.1080/07362994.2016.1180995. |
| [5] |
M. K. Ghosh and S. Saha,
Risk-sensitive control of continuous time Markov chains, Stochastics, 86 (2014), 655-675.
doi: 10.1080/17442508.2013.872644. |
| [6] |
X. Guo,
Continuous-time Markov decision processes with discounted rewards: The case of Polish spaces, Math. Oper. Res., 32 (2007), 73-87.
doi: 10.1287/moor.1060.0210. |
| [7] |
X. Guo and O. Hernández-Lerma,
Nonzero-sum games for continuous-time Markov chains with unbounded discounted payoffs, J. Appl. Probab., 42 (2005), 303-320.
doi: 10.1239/jap/1118777172. |
| [8] |
X. Guo and O. Hernández-Lerma,
Zero-sum games for continuous-time jump Markov processes in Polish spaces: Discounted payoffs, Adv. in Appl. Probab., 39 (2007), 645-668.
doi: 10.1017/S0001867800001981. |
| [9] |
X. Guo and O. Hernández-Lerma,
Zero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates, J. Appl. Probab., 40 (2003), 327-345.
doi: 10.1017/S0021900200019331. |
| [10] |
X. Guo and Z.-W. Liao,
Risk-sensitive discounted continuous-time Markov decision processes with unbounded rates, SIAM J. Control Optim., 57 (2019), 3857-3883.
doi: 10.1137/18M1222016. |
| [11] |
X. Guo and Y. Zhang,
On risk-sensitive piecewise deterministic Markov decision processes, Appl. Math. Optim., 81 (2020), 685-710.
doi: 10.1007/s00245-018-9485-x. |
| [12] |
O. Hernández-Lerma and J. B. Lasserre, Further Topics on Discrete-Time Markov Control Processes, Applications of Mathematics (New York), 42, Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-0561-6. |
| [13] |
A. S. Nowak, Notes on risk-sensitive Nash equilibria, in Advances in Dynamic Games, Ann. Internat. Soc. Dynam. Games, 7, Birkhäuser Boston, Boston, MA, 2005, 95–109.
doi: 10.1007/0-8176-4429-6_5. |
| [14] |
C. Pal and S. Pradhan,
Risk sensitive control of pure jump processes on a general state space, Stochastics, 91 (2019), 155-174.
doi: 10.1080/17442508.2018.1521413. |
| [15] |
K. Suresh Kumar and C. Pal,
Risk-sensitive control of pure jump process on countable space with near monotone cost, Appl. Math. Optim., 68 (2013), 311-331.
doi: 10.1007/s00245-013-9208-2. |
| [16] |
K. Suresh Kumar and C. Pal,
Risk-sensitive ergodic control of continuous time Markov processes with denumerable state space, Stoch. Anal. Appl., 33 (2015), 863-881.
doi: 10.1080/07362994.2015.1050674. |
| [17] |
Q. Wei,
Zero-sum games for continuous-time Markov jump processes with risk-sensitive finite-horizon cost criterion, Oper. Res. Lett., 46 (2018), 69-75.
doi: 10.1016/j.orl.2017.11.008. |
| [18] |
Q. Wei and X. Chen,
Stochastic games for continuous-time jump processes under finite-horizon payoff criterion, Appl. Math. Optim., 74 (2016), 273-301.
doi: 10.1007/s00245-015-9314-4. |
| [19] |
P. Whittle, Risk-Sensitive Optimal Control, Wiley-Interscience Series in Systems and Optimization, John Wiley & Sons, Ltd., Chichester, 1990. |
| [20] |
W. Zhang,
Continuous-time constrained stochastic games under the discounted cost criteria, Appl. Math. Optim., 77 (2018), 275-296.
doi: 10.1007/s00245-016-9374-0. |
| [21] |
Y. Zhang,
Continuous-time Markov decision processes with exponential utility, SIAM J. Control Optim., 55 (2017), 2636-2660.
doi: 10.1137/16M1086261. |
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