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December  2018, 7(4): 587-597. doi: 10.3934/eect.2018028

Some partially observed multi-agent linear exponential quadratic stochastic differential games

Tyrone E. Duncan

Mathematics Department, Snow Hall, 1460 Jayhawk Blvd, Lawrence, KS 66045, USA

Received  January 2018 Revised  August 2018 Published  September 2018

Fund Project: The author was supported by NSF grant DMS 1411412, AFOSR grant FA9550-17-1-0073, and ARO grant W911NF-14-10390.

Some multi-agent stochastic differential games described by a stochastic linear system driven by a Brownian motion and having an exponential quadratic payoff for the agents are formulated and solved. The agents have either complete observations or partial observations of the system state. The agents act independently of one another and the explicit optimal feedback control strategies form a Nash equilibrium. In the partially observed problem the observations are the same for all agents which occurs in broadcast situations. The optimal control strategies and optimal payoffs are given explicitly. The method of solution for both problems does not require solving either Hamilton-Jacobi-Isaacs equations or backward stochastic differential equations.

Keywords: Stochastic differential games, multi-agent games, partially observed games, risk sensitive games, exponential quadratic payoffs.
Mathematics Subject Classification: Primary: 91A15, 49N70; Secondary: 91A25.
Citation: Tyrone E. Duncan. Some partially observed multi-agent linear exponential quadratic stochastic differential games. Evolution Equations & Control Theory, 2018, 7 (4) : 587-597. doi: 10.3934/eect.2018028
References:
[1]

M. Bardi and F. S. Priuli, Linear-quadratic n-person and mean-field games with ergodic cost, SIAM J. Control Optim., 52 (2014), 3022-3052.  doi: 10.1137/140951795.  Google Scholar

[2]

T. Basar and P. Bernhard, H∞-Optimal Control and Related Minimax Design Problems, Birkhauser, Boston, 1995. doi: 10.1007/978-0-8176-4757-5.  Google Scholar

[3]

T. Basar and G. Olsder, Dynamic Noncooperative Game Theory, SIAM, 2nd Ed. 1999.  Google Scholar

[4]

A. Bensoussan and J. H. van Schuppen, Optimal control of partially observable stochastic systems with an exponential-of-integral performance index, SIAM J. Control Optim., 23 (1995), 599-613.  doi: 10.1137/0323038.  Google Scholar

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R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations, SIAM J. Control Optim., 47 (2008), 444-475.  doi: 10.1137/060671954.  Google Scholar

[6]

T. E. Duncan, Evaluation of likelihood functions, Information and Control, 13 (1968), 62-74.  doi: 10.1016/S0019-9958(68)90795-X.  Google Scholar

[7]

T. E. Duncan, Linear-exponential-quadratic Gaussian control, IEEE Trans. Autom. Control, 58 (2013), 2910-2911.  doi: 10.1109/TAC.2013.2257610.  Google Scholar

[8]

T. E. Duncan, Some linear-quadratic stochastic differential games for equations in a Hilbert space with fractional Brownian motions, Discrete Cont. Dyn. Systems Ser. A, 35 (2015), 5435-5445.  doi: 10.3934/dcds.2015.35.5435.  Google Scholar

[9]

T. E. Duncan, Linear exponential quadratic stochastic differential games, IEEE Trans. Autom. Control, 61 (2016), 2550-2552.  doi: 10.1109/TAC.2015.2510983.  Google Scholar

[10]

T. E. Duncan and B. Pasik-Duncan, A solvable stochastic differential game in the two-sphere, Proc. 52nd IEEE Conf. Decision and Control, 7833–7837, Firenze, 2013. Google Scholar

[11]

T. E. Duncan and B. Pasik-Duncan, Some results on optimal control for a partially observed linear stochastic system with an exponential quadratic cost, Proc. IFAC World Congress, Cape Town, 47 (2014), 8695-8698. doi: 10.3182/20140824-6-ZA-1003.00522.  Google Scholar

[12]

R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models, Springer, 1995.  Google Scholar

[13]

W. H. Fleming and D. Hernandez-Hernandez, On the value of stochastic differential games, Commun. Stoch. Anal., 5 (2011), 341-351.   Google Scholar

[14]

W. H. Fleming and P. E. Souganidis, On the existence of value functions of two player, zero sum stochastic differential games, Indiana Math. J., 38 (1989), 293-314.  doi: 10.1512/iumj.1989.38.38015.  Google Scholar

[15]

S. Hamadene, J. P. Lepeltier and S. Peng, BSDEs with continuous coefficients and stochastic differential games, Backward Stochastic Differential Equations (N. El Karoui et al., eds.), Pitman Res. Notes Math, 364 (1997), 115-128.  Google Scholar

[16]

R. Isaacs, Differential Games, J. Wiley, New York 1965. Google Scholar

[17]

D. H. Jacobson, Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games, IEEE Trans. Autom. Control, AC-18 (1973), 124-131.   Google Scholar

[18]

M. L. Kleptsyna, A. Le Breton and M. Viot, On the linear-exponential filtering problem for general Gaussian processes, SIAM J. control Optim., 47 (2008), 2886-2911.  doi: 10.1137/070705908.  Google Scholar

[19]

J. B. Moore, R. J. Elliott and S. Dey, Risk sensitive generalizations of minimum variance estimation and control, J. Math. Syst., Estimation, Contr., (electronic), 7 (1997), 15 pp.  Google Scholar

[20]

J. Nash, Non-cooperative games, Ann. Math., 54 (1951), 286-295.  doi: 10.2307/1969529.  Google Scholar

[21]

J. L. Speyer, J. Deyst and D. H. Jacobson, Optimization of stochastic linear systems with additive measurement and process noise using exponential performance criteria, IEEE Trans. Autom. Control, AC-19 (1974), 358-366.   Google Scholar

show all references

References:
[1]

M. Bardi and F. S. Priuli, Linear-quadratic n-person and mean-field games with ergodic cost, SIAM J. Control Optim., 52 (2014), 3022-3052.  doi: 10.1137/140951795.  Google Scholar

[2]

T. Basar and P. Bernhard, H∞-Optimal Control and Related Minimax Design Problems, Birkhauser, Boston, 1995. doi: 10.1007/978-0-8176-4757-5.  Google Scholar

[3]

T. Basar and G. Olsder, Dynamic Noncooperative Game Theory, SIAM, 2nd Ed. 1999.  Google Scholar

[4]

A. Bensoussan and J. H. van Schuppen, Optimal control of partially observable stochastic systems with an exponential-of-integral performance index, SIAM J. Control Optim., 23 (1995), 599-613.  doi: 10.1137/0323038.  Google Scholar

[5]

R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations, SIAM J. Control Optim., 47 (2008), 444-475.  doi: 10.1137/060671954.  Google Scholar

[6]

T. E. Duncan, Evaluation of likelihood functions, Information and Control, 13 (1968), 62-74.  doi: 10.1016/S0019-9958(68)90795-X.  Google Scholar

[7]

T. E. Duncan, Linear-exponential-quadratic Gaussian control, IEEE Trans. Autom. Control, 58 (2013), 2910-2911.  doi: 10.1109/TAC.2013.2257610.  Google Scholar

[8]

T. E. Duncan, Some linear-quadratic stochastic differential games for equations in a Hilbert space with fractional Brownian motions, Discrete Cont. Dyn. Systems Ser. A, 35 (2015), 5435-5445.  doi: 10.3934/dcds.2015.35.5435.  Google Scholar

[9]

T. E. Duncan, Linear exponential quadratic stochastic differential games, IEEE Trans. Autom. Control, 61 (2016), 2550-2552.  doi: 10.1109/TAC.2015.2510983.  Google Scholar

[10]

T. E. Duncan and B. Pasik-Duncan, A solvable stochastic differential game in the two-sphere, Proc. 52nd IEEE Conf. Decision and Control, 7833–7837, Firenze, 2013. Google Scholar

[11]

T. E. Duncan and B. Pasik-Duncan, Some results on optimal control for a partially observed linear stochastic system with an exponential quadratic cost, Proc. IFAC World Congress, Cape Town, 47 (2014), 8695-8698. doi: 10.3182/20140824-6-ZA-1003.00522.  Google Scholar

[12]

R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models, Springer, 1995.  Google Scholar

[13]

W. H. Fleming and D. Hernandez-Hernandez, On the value of stochastic differential games, Commun. Stoch. Anal., 5 (2011), 341-351.   Google Scholar

[14]

W. H. Fleming and P. E. Souganidis, On the existence of value functions of two player, zero sum stochastic differential games, Indiana Math. J., 38 (1989), 293-314.  doi: 10.1512/iumj.1989.38.38015.  Google Scholar

[15]

S. Hamadene, J. P. Lepeltier and S. Peng, BSDEs with continuous coefficients and stochastic differential games, Backward Stochastic Differential Equations (N. El Karoui et al., eds.), Pitman Res. Notes Math, 364 (1997), 115-128.  Google Scholar

[16]

R. Isaacs, Differential Games, J. Wiley, New York 1965. Google Scholar

[17]

D. H. Jacobson, Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games, IEEE Trans. Autom. Control, AC-18 (1973), 124-131.   Google Scholar

[18]

M. L. Kleptsyna, A. Le Breton and M. Viot, On the linear-exponential filtering problem for general Gaussian processes, SIAM J. control Optim., 47 (2008), 2886-2911.  doi: 10.1137/070705908.  Google Scholar

[19]

J. B. Moore, R. J. Elliott and S. Dey, Risk sensitive generalizations of minimum variance estimation and control, J. Math. Syst., Estimation, Contr., (electronic), 7 (1997), 15 pp.  Google Scholar

[20]

J. Nash, Non-cooperative games, Ann. Math., 54 (1951), 286-295.  doi: 10.2307/1969529.  Google Scholar

[21]

J. L. Speyer, J. Deyst and D. H. Jacobson, Optimization of stochastic linear systems with additive measurement and process noise using exponential performance criteria, IEEE Trans. Autom. Control, AC-19 (1974), 358-366.   Google Scholar

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