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doi: 10.3934/jdg.2021020
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Zero-sum games for pure jump processes with risk-sensitive discounted cost criteria

Chandan Pal 1,, and  Somnath Pradhan 2,

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

* Corresponding author: Chandan Pal

Received  April 2021 Revised  June 2021 Early access July 2021

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.

Keywords: Pure jump process, Markov strategy, value of the game, saddle point equilibrium, HJI equation.
Mathematics Subject Classification: Primary: 91A15, 91A25; Secondary: 60J75.
Citation: Chandan Pal, Somnath Pradhan. Zero-sum games for pure jump processes with risk-sensitive discounted cost criteria. Journal of Dynamics & Games, doi: 10.3934/jdg.2021020
References:
[1] R. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

P. Whittle, Risk-Sensitive Optimal Control, Wiley-Interscience Series in Systems and Optimization, John Wiley & Sons, Ltd., Chichester, 1990.  Google Scholar

[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.  Google Scholar

[21]

Y. Zhang, Continuous-time Markov decision processes with exponential utility, SIAM J. Control Optim., 55 (2017), 2636-2660.  doi: 10.1137/16M1086261.  Google Scholar

show all references

References:
[1] R. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

P. Whittle, Risk-Sensitive Optimal Control, Wiley-Interscience Series in Systems and Optimization, John Wiley & Sons, Ltd., Chichester, 1990.  Google Scholar

[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.  Google Scholar

[21]

Y. Zhang, Continuous-time Markov decision processes with exponential utility, SIAM J. Control Optim., 55 (2017), 2636-2660.  doi: 10.1137/16M1086261.  Google Scholar

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