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doi: 10.3934/mcrf.2020028

Nonzero-sum differential game of backward doubly stochastic systems with delay and applications

1. 

Institute for Financial Studies and School of Mathematics, Shandong University, Jinan 250100, China

2. 

School of Mathematics and Quantitative Economics, and Shandong Key Laboratory of Blockchain Finance, Shandong University of Finance and Economics, Jinan 250014, China

* Corresponding author: Yufeng Shi

Received  August 2019 Revised  March 2020 Published  June 2020

Fund Project: This work was supported by National Key R & D Program of China (2018YFA0703900), National Natural Science Foundation of China (11871309, 11671229, 11371226, 11301298), Natural Science Foundation of Shandong Province of China (ZR2019MA013), the Special Funds of Taishan Scholar Project (Grant No. tsqn20161041), and Fostering Project of Dominant Discipline and Talent Team of Shandong Province Higher Education Institutions

This paper is concerned with a kind of nonzero-sum differential game of backward doubly stochastic system with delay, in which the state dynamics follows a delayed backward doubly stochastic differential equation (SDE). To deal with the above game problem, it is natural to involve the adjoint equation, which is a kind of anticipated forward doubly SDE. We give the existence and uniqueness of solutions to delayed backward doubly SDE and anticipated forward doubly SDE. We establish a necessary condition in the form of maximum principle with Pontryagin's type for open-loop Nash equilibrium point of this type of game, and then give a verification theorem which is a sufficient condition for Nash equilibrium point. The theoretical results are applied to study a nonzero-sum differential game of linear-quadratic backward doubly stochastic system with delay.

Citation: Qingfeng Zhu, Yufeng Shi. Nonzero-sum differential game of backward doubly stochastic systems with delay and applications. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020028
References:
[1]

K. BahlaliR. GattB. Mansouri and A. Mtiraoui, Backward doubly SDEs and SPDEs with superlinear growth generators, Stoch. Dyn., 17 (2017), 1-31.  doi: 10.1142/S0219493717500095.  Google Scholar

[2]

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J. Shi, G. Wang and J. Xiong, Linear-quadratic stochastic Stackelberg differential game with asymmetric information, Sci. China. Inf. Sci., 60 (2017), 092202. doi: 10.1007/s11432-016-0654-y.  Google Scholar

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G. Wang and Z. Yu, A Pontryagin's maximum principle for nonzero sum differential games of BSDEs with applications, IEEE Trans. Autom. Control, 55 (2010), 1742-1747.  doi: 10.1109/TAC.2010.2048052.  Google Scholar

[21]

G. Wang and Z. Yu, A partial information non-zero sum differential game of backward stochastic differential equations with applications, Automatica J. IFAC, 48 (2012), 342-352.  doi: 10.1016/j.automatica.2011.11.010.  Google Scholar

[22]

T. Wang and Y. Shi, Linear quadratic stochastic integral games and related topics, Sci. China. Math., 58 (2015), 2405-2420.  doi: 10.1007/s11425-015-5026-0.  Google Scholar

[23]

Q. Wei and Z. Yu, Time-inconsistent recursive zero-sum stochastic differential games, Math. Control Relat. Fields, 8 (2018), 1051-1079.  doi: 10.3934/mcrf.2018045.  Google Scholar

[24]

Z. Wu and F. Zhang, BDSDEs with locally monotone coefficients and Sobolev solutions for SPDEs, J. Differential Equations, 251 (2011), 759-784.  doi: 10.1016/j.jde.2011.05.017.  Google Scholar

[25]

J. Xu and Y. Han, Stochastic maximum principle for delayed backward doubly stochastic control systems, J. Nonlinear Sci. Appl., 10 (2017), 215-226.  doi: 10.22436/jnsa.010.01.21.  Google Scholar

[26]

J. Xu, Stochastic maximum principle for delayed backward doubly stochastic control systems and their applications, Int. J. Control, (2018). doi: 10.1080/00207179.2018.1508850.  Google Scholar

[27]

Z. Yu and S. Ji, Linear-quadratic non-zero sum differential game of backward stochstic differential equations, in Proceedings of the 27th Chinese Control Conference, Kunming, Yunnan, (2008), 562–566. Google Scholar

[28]

L. Zhang and Y. Shi, Maximum principle for forward-backward doubly stochastic control systems and applications, ESAIM Control Optim. Calc. Var., 17 (2011), 1174-1197.  doi: 10.1051/cocv/2010042.  Google Scholar

[29]

Q. Zhang and H. Zhao, Stationary solutions of SPDEs and infinite horizon BDSDEs, J. Funct. Anal., 252 (2007), 171-219.  doi: 10.1016/j.jfa.2007.06.019.  Google Scholar

[30]

Q. Zhang and H. Zhao, Stationary solutions of SPDEs and infinite horizon BDSDEs under non-Lipschitz coefficients, J. Differential Equations, 248 (2010), 953-991.  doi: 10.1016/j.jde.2009.12.013.  Google Scholar

[31]

Q. Zhang and H. Zhao, SPDEs with polynomial growth coefficients and the Malliavin calculus method, Stochastic Process. Appl., 123 (2013), 2228-2271.  doi: 10.1016/j.spa.2013.02.004.  Google Scholar

[32]

Q. Zhang and H. Zhao, Backward doubly stochastic differential equations with polynomial growth coefficients, Discrete Contin. Dyn. Syst., 35 (2015), 5285-5315.  doi: 10.3934/dcds.2015.35.5285.  Google Scholar

[33]

Q. ZhuY. Shi and X. Gong, Solutions to general forward-backward doubly stochastic differential equations, Appl. Math. Mech., 30 (2009), 517-526.  doi: 10.1007/s10483-009-0412-x.  Google Scholar

[34]

Q. Zhu and Y. Shi, Forward-backward doubly stochastic differential equations and related stochastic partial differential equations, Sci. China Math., 55 (2012), 2517-2534.  doi: 10.1007/s11425-012-4411-1.  Google Scholar

[35]

Q. Zhu and Y. Shi, Optimal control of backward doubly stochastic systems with partial information, IEEE Trans. Automat. Control, 60 (2015), 173-178.  doi: 10.1109/TAC.2014.2322212.  Google Scholar

show all references

References:
[1]

K. BahlaliR. GattB. Mansouri and A. Mtiraoui, Backward doubly SDEs and SPDEs with superlinear growth generators, Stoch. Dyn., 17 (2017), 1-31.  doi: 10.1142/S0219493717500095.  Google Scholar

[2]

V. Bally and A. Matoussi, Weak solutions for SPDEs and backward doubly stochastic differential equations, J. Theoret. Probab., 14 (2001), 125-164.  doi: 10.1023/A:1007825232513.  Google Scholar

[3]

F. Biagini and B. Øksendal, Minimal variance hedging for insider trading, Int. J. Theor. Appl. Finance, 9 (2006), 1351-1375.  doi: 10.1142/S0219024906003998.  Google Scholar

[4]

L. Campi, Some results on quadratic hedging with insider trading, Stochastics, 77 (2005), 327-348.  doi: 10.1080/17442500500183503.  Google Scholar

[5]

L. Chen and Z. Wu, Maximum principle for the stochastic optimal control problem with delay and application, Automatica J. IFAC, 46 (2010), 1074-1080.  doi: 10.1016/j.automatica.2010.03.005.  Google Scholar

[6]

L. Chen and Z. Wu, A type of generalized forward-backward stochastic differential equations and applications, Chin. Ann. Math. Ser. B, 32 (2011), 279-292.  doi: 10.1007/s11401-011-0631-x.  Google Scholar

[7]

L. Chen and Z. Yu, Maximum principle for nonzero-sum stochastic differential game with delays, IEEE Trans. Automat. Control, 60 (2015), 1422-1426.  doi: 10.1109/TAC.2014.2352731.  Google Scholar

[8]

K. DuJ. Huang and Z. Wu, Linear quadratic mean-field-game of backward stochastic differential systems, Math. Control Relat. Fields, 8 (2018), 653-678.  doi: 10.3934/mcrf.2018028.  Google Scholar

[9]

Y. HanS. Peng and Z. Wu, Maximum principle for backward doubly stochastic control systems with applications, SIAM J. Control Optim., 48 (2010), 4224-4241.  doi: 10.1137/080743561.  Google Scholar

[10]

L. Hu and Y. Ren, Stochastic PDIEs with nonlinear Neumann boundary conditions and generalized backward doubly stochastic differential equations driven by Lévy processes, J. Comput. Appl. Math., 229 (2009), 230-239.  doi: 10.1016/j.cam.2008.10.027.  Google Scholar

[11]

A. MatoussiL. Piozin and A. Popier, Stochastic partial differential equations with singular terminal condition, Stochastic Process. Appl., 127 (2017), 831-876.  doi: 10.1016/j.spa.2016.07.002.  Google Scholar

[12]

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

[13]

E. Pardoux and S. Peng, Backward doubly stochastic differential equations and systems of quasilinear parabolic SPDEs, Probab. Theory Related Fields, 98 (1994), 209-227.  doi: 10.1007/BF01192514.  Google Scholar

[14]

S. Peng and Y. Shi, A type of time-symmetric forward-backward stochastic differential equations, C. R. Acad. Sci. Paris, 336 (2003), 773-778.  doi: 10.1016/S1631-073X(03)00183-3.  Google Scholar

[15]

Y. RenA. Lin and L. Hu, Stochastic PDIEs and backward doubly stochastic differential equations driven by Lévy processes, J. Comput. Appl. Math., 223 (2009), 901-907.  doi: 10.1016/j.cam.2008.03.008.  Google Scholar

[16]

J. Shi and G. Wang, A nonzero sum differential game of BSDE with time-delayed generator and applications, IEEE Trans. Automat. Control, 61 (2016), 1959-1964.  doi: 10.1109/TAC.2015.2480335.  Google Scholar

[17]

J. Shi, G. Wang and J. Xiong, Linear-quadratic stochastic Stackelberg differential game with asymmetric information, Sci. China. Inf. Sci., 60 (2017), 092202. doi: 10.1007/s11432-016-0654-y.  Google Scholar

[18]

Y. Shi and Q. Zhu, Partially observed optimal control of forward-backward doubly stochastic systems, ESAIM Control Optim. Calc. Var., 19 (2013), 828-843.  doi: 10.1051/cocv/2012035.  Google Scholar

[19] J. Von Neumann and O. Morgenstern, The Theory of Games and Economic Behavior, Princeton University Press, Princeton, New Jersey, 1944.   Google Scholar
[20]

G. Wang and Z. Yu, A Pontryagin's maximum principle for nonzero sum differential games of BSDEs with applications, IEEE Trans. Autom. Control, 55 (2010), 1742-1747.  doi: 10.1109/TAC.2010.2048052.  Google Scholar

[21]

G. Wang and Z. Yu, A partial information non-zero sum differential game of backward stochastic differential equations with applications, Automatica J. IFAC, 48 (2012), 342-352.  doi: 10.1016/j.automatica.2011.11.010.  Google Scholar

[22]

T. Wang and Y. Shi, Linear quadratic stochastic integral games and related topics, Sci. China. Math., 58 (2015), 2405-2420.  doi: 10.1007/s11425-015-5026-0.  Google Scholar

[23]

Q. Wei and Z. Yu, Time-inconsistent recursive zero-sum stochastic differential games, Math. Control Relat. Fields, 8 (2018), 1051-1079.  doi: 10.3934/mcrf.2018045.  Google Scholar

[24]

Z. Wu and F. Zhang, BDSDEs with locally monotone coefficients and Sobolev solutions for SPDEs, J. Differential Equations, 251 (2011), 759-784.  doi: 10.1016/j.jde.2011.05.017.  Google Scholar

[25]

J. Xu and Y. Han, Stochastic maximum principle for delayed backward doubly stochastic control systems, J. Nonlinear Sci. Appl., 10 (2017), 215-226.  doi: 10.22436/jnsa.010.01.21.  Google Scholar

[26]

J. Xu, Stochastic maximum principle for delayed backward doubly stochastic control systems and their applications, Int. J. Control, (2018). doi: 10.1080/00207179.2018.1508850.  Google Scholar

[27]

Z. Yu and S. Ji, Linear-quadratic non-zero sum differential game of backward stochstic differential equations, in Proceedings of the 27th Chinese Control Conference, Kunming, Yunnan, (2008), 562–566. Google Scholar

[28]

L. Zhang and Y. Shi, Maximum principle for forward-backward doubly stochastic control systems and applications, ESAIM Control Optim. Calc. Var., 17 (2011), 1174-1197.  doi: 10.1051/cocv/2010042.  Google Scholar

[29]

Q. Zhang and H. Zhao, Stationary solutions of SPDEs and infinite horizon BDSDEs, J. Funct. Anal., 252 (2007), 171-219.  doi: 10.1016/j.jfa.2007.06.019.  Google Scholar

[30]

Q. Zhang and H. Zhao, Stationary solutions of SPDEs and infinite horizon BDSDEs under non-Lipschitz coefficients, J. Differential Equations, 248 (2010), 953-991.  doi: 10.1016/j.jde.2009.12.013.  Google Scholar

[31]

Q. Zhang and H. Zhao, SPDEs with polynomial growth coefficients and the Malliavin calculus method, Stochastic Process. Appl., 123 (2013), 2228-2271.  doi: 10.1016/j.spa.2013.02.004.  Google Scholar

[32]

Q. Zhang and H. Zhao, Backward doubly stochastic differential equations with polynomial growth coefficients, Discrete Contin. Dyn. Syst., 35 (2015), 5285-5315.  doi: 10.3934/dcds.2015.35.5285.  Google Scholar

[33]

Q. ZhuY. Shi and X. Gong, Solutions to general forward-backward doubly stochastic differential equations, Appl. Math. Mech., 30 (2009), 517-526.  doi: 10.1007/s10483-009-0412-x.  Google Scholar

[34]

Q. Zhu and Y. Shi, Forward-backward doubly stochastic differential equations and related stochastic partial differential equations, Sci. China Math., 55 (2012), 2517-2534.  doi: 10.1007/s11425-012-4411-1.  Google Scholar

[35]

Q. Zhu and Y. Shi, Optimal control of backward doubly stochastic systems with partial information, IEEE Trans. Automat. Control, 60 (2015), 173-178.  doi: 10.1109/TAC.2014.2322212.  Google Scholar

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