# American Institute of Mathematical Sciences

March  2021, 11(1): 73-94. 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  March 2021 Early access  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 and Related Fields, 2021, 11 (1) : 73-94. doi: 10.3934/mcrf.2020028
##### References:
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Control, 61 (2016), 1959-1964.  doi: 10.1109/TAC.2015.2480335. [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. [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. [19] J. Von Neumann and O. Morgenstern, The Theory of Games and Economic Behavior, Princeton University Press, Princeton, New Jersey, 1944. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [33] Q. Zhu, Y. 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. [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. [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.

show all references

##### References:
 [1] K. Bahlali, R. Gatt, B. Mansouri and A. Mtiraoui, Backward doubly SDEs and SPDEs with superlinear growth generators, Stoch. Dyn., 17 (2017), 1-31.  doi: 10.1142/S0219493717500095. [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. [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. [4] L. Campi, Some results on quadratic hedging with insider trading, Stochastics, 77 (2005), 327-348.  doi: 10.1080/17442500500183503. [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. [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. [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. [8] K. Du, J. 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. [9] Y. Han, S. 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. [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. [11] A. Matoussi, L. 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. [12] J. Nash, Non-cooperative games, Ann. of Math., 54 (1951), 286-295.  doi: 10.2307/1969529. [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. [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. [15] Y. Ren, A. 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. [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. [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. [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. [19] J. Von Neumann and O. Morgenstern, The Theory of Games and Economic Behavior, Princeton University Press, Princeton, New Jersey, 1944. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [33] Q. Zhu, Y. 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. [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. [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.
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