December  2021, 11(4): 797-828. doi: 10.3934/mcrf.2020047

A stackelberg game of backward stochastic differential equations with partial information

School of Mathematics, Shandong University, Jinan 250100, China

* Corresponding author: Jingtao Shi

Received  December 2019 Revised  June 2020 Published  December 2021 Early access  November 2020

Fund Project: This work is financially supported by National Key R & D Program of China (2018YFB1305400) and National Natural Science Foundations of China (11971266, 11831010, 11571205)

This paper is concerned with a Stackelberg game of backward stochastic differential equations (BSDEs) with partial information, where the information of the follower is a sub-$ \sigma $-algebra of that of the leader. Necessary and sufficient conditions of the optimality for the follower and the leader are first given for the general problem, by the partial information stochastic maximum principles of BSDEs and forward-backward stochastic differential equations (FBSDEs), respectively. Then a linear-quadratic (LQ) Stackelberg game of BSDEs with partial information is investigated. The state estimate feedback representation for the optimal control of the follower is first given via two Riccati equations. Then the leader's problem is formulated as an optimal control problem of FBSDE. Four high-dimensional Riccati equations are introduced to represent the state estimate feedback for the optimal control of the leader. Theoretic results are applied to a pension fund management problem of two players in the financial market.

Citation: Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control and Related Fields, 2021, 11 (4) : 797-828. doi: 10.3934/mcrf.2020047
References:
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A. Aurell, Mean-field type games between two players driven by backward stochastic differential equations, Games, 9 (2018), Paper No. 88, 26 pp. doi: 10.3390/g9040088.

[2]

A. Bagchi and T. Başar, Stackelberg strategies in linear-quadratic stochastic differential games, J. Optim. Theory Appl., 35 (1981), 443-464.  doi: 10.1007/BF00934911.

[3]

A. BensoussanS. K. Chen and S. P. Sethi, The maximum principle for global solutions of stochastic Stackelberg differential games, SIAM J. Control Optim., 53 (2015), 1956-1981.  doi: 10.1137/140958906.

[4]

J. Bismut, An introductory approach to duality in optimal stochastic control, SIAM Rev., 20 (1978), 62-78.  doi: 10.1137/1020004.

[5]

R. BuckdahnB. DjehicheJ. Li and S. G. Peng, Mean-field backward stochastic differential equations: A limit approach, Ann. Probab., 37 (2009), 1524-1565.  doi: 10.1214/08-AOP442.

[6]

S. P. Chen and X. Y. Zhou, Stochastic linear quadratic regulators with indefinite control weight costs. II, SIAM J. Control Optim., 39 (2000), 1065-1081.  doi: 10.1137/S0363012998346578.

[7]

M. Di GiacintoS. Federico and F. Gozzi, Pension funds with a minimum guarantee: A stochastic control approach, Finance Stoch., 15 (2011), 297-342.  doi: 10.1007/s00780-010-0127-7.

[8]

N. G. Dokuchaev and X. Y. Zhou, Stochastic controls with terminal contingent conditions, J. Math. Anal. Appl., 238 (1999), 143-165.  doi: 10.1006/jmaa.1999.6515.

[9]

K. DuJ. H. Huang and Z. Wu, Linear quadratic mean-field-game of backward stochastic differential systems, Math. Control Relat. Fields, 8 (2019), 653-678.  doi: 10.3934/mcrf.2018028.

[10]

K. Du and Z. Wu, Linear-quadratic Stackelberg game for mean-field backward stochastic differential system and application, Math. Probl. Eng., Vol. 2019, Art. Id. 1798585, 17 pp. doi: 10.1155/2019/1798585.

[11]

D. Duffie and L. G. Epstein, Stochastic Differential Utility, Econometrica, 60 (1992), 353-394.  doi: 10.2307/2951600.

[12]

N. El KarouiS. G. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

[13]

S. Hamadene and J. P. Lepeltier, Zero-sum stochastic differential games and backward equations, Systems Control Lett., 24 (1995), 259-263.  doi: 10.1016/0167-6911(94)00011-J.

[14]

J. H. HuangG. C. Wang and J. Xiong, A maximum principle for partial information backward stochastic control problems with applications, SIAM J. Control Optim., 48 (2009), 2106-2117.  doi: 10.1137/080738465.

[15]

J. H. HuangS. J. Wang and Z. Wu, Backward mean-field Linear-Quadratic-Gaussian (LQG) games: Full and partial information, IEEE Trans. Automat. Control, 61 (2016), 3784-3796.  doi: 10.1109/TAC.2016.2519501.

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P. Y. Huang and G. C. Wang, A non-zero sum differential game of mean-field backward stochastic differential equation, in Proc. 2017 Chinese Automation Congress, Jinan, 2017, 4827-4831.

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P. Y. Huang, G. C. Wang and H. J. Zhang, An asymmetric information non-zero sum differential game of mean-field backward stochastic differential equation with applications, Adv. Differ. Equ., 2019, Paper No. 236, 25 pp. doi: 10.1186/s13662-019-2166-5.

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M. Kohlmann and X. Y. Zhou, Relationship between backward stochastic differential equations and stochastic controls: A linear-quadratic approach, SIAM J. Control Optim., 38 (2000), 1392-1407.  doi: 10.1137/S036301299834973X.

[19]

N. Li and Z. Y. Yu, Forward-backward stochastic differential equations and linear-quadratic generalized Stackelberg games, SIAM J. Control Optim., 56 (2018), 4148-4180.  doi: 10.1137/17M1158392.

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X. LiJ. R. Sun and J. Xiong, Linear quadratic optimal control problems for mean-field backward stochastic differential equations, Appl. Math. Optim., 80 (2019), 223-250.  doi: 10.1007/s00245-017-9464-7.

[21]

A. E. B. Lim and X. Y. Zhou, Linear-quadratic control of backward stochastic differential equations, SIAM J. Control Optim., 40 (2001), 450-474.  doi: 10.1137/S0363012900374737.

[22]

Y. N. LinX. S. Jiang and W. H. Zhang, An open-loop Stackelberg strategy for the linear quadratic mean-field stochastic differential game, IEEE Trans. Automat. Control, 64 (2019), 97-110.  doi: 10.1109/TAC.2018.2814959.

[23]

H. P. Ma and B. Liu, Infinite horizon optimal control problem of mean-field backward stochastic delay differential equation under partial information, Eur. J. Control, 36 (2017), 43-50.  doi: 10.1016/j.ejcon.2017.04.001.

[24]

J. Moon and T. Başar, Linear quadratic mean field Stackelberg differential games, Automatica J. IFAC, 97 (2018), 200-213.  doi: 10.1016/j.automatica.2018.08.008.

[25]

B. ØksendalL. Sandal and J. Ubøe, Stochastic Stackelberg equilibria with applications to time dependent newsvendor models, J. Econom. Dynam. Control, 37 (2013), 1284-1299.  doi: 10.1016/j.jedc.2013.02.010.

[26]

E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14 (1990), 55-61.  doi: 10.1016/0167-6911(90)90082-6.

[27]

E. Pardoux and A. Răşcanu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Springer, Cham, 2014. doi: 10.1007/978-3-319-05714-9.

[28]

S. G. Peng, A generalized dynamic programming principle and Hamilton-Jacobi-Bellmen equation, Stochastics Stochastics Rep., 38 (1992), 119-134.  doi: 10.1080/17442509208833749.

[29]

S. G. Peng, Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim., 27 (1993), 125-144.  doi: 10.1007/BF01195978.

[30]

J. T. Shi, Optimal control of backward stochastic differential equations with time delayed generators. In Proc. 30th Chinese Control Conference, Yantai, China, 2011, 1285-1289.

[31]

J. T. Shi, Sufficient conditions of optimality for mean-field stochastic control problems, In Proc. 12th International Conference on Automation, Robots, Control and Vision, Guangzhou, China, 2012,747-752. doi: 10.1109/ICARCV.2012.6485251.

[32]

J. T. Shi and G. C. Wang, A non-zero 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.

[33]

J. T. ShiG. C. Wang and J. Xiong, A Leader-follower stochastic differential game with asymmetric information and applications, Automatica J. IFAC, 63 (2016), 60-73.  doi: 10.1016/j.automatica.2015.10.011.

[34]

J. T. Shi, G. C. Wang and J. Xiong, Linear-quadratic stochastic Stackelberg differential game with asymmetric information, Sci. China Infor. Sci., 60 (2017), 092202, 1-15.

[35]

J. T. Shi, G. C. Wang and J. Xiong, Stochastic linear quadratic Stackelberg differential game with overlapping information, ESAIM: Control, Optim. Calcu. Varia., 26 (2020), Art. No. 83, 38 pp. doi: 10.1051/cocv/2020006.

[36]

M. Simaan and J. B. Cruz Jr., On the Stackelberg game strategy in nonzero-sum games, J. Optim. Theory Appl., 11 (1973), 533-555.  doi: 10.1007/BF00935665.

[37]

H. von Stackelberg, Marktform und Gleichgewicht, Springer, Vienna, 1934.

[38]

G. C. WangH. Xiao and G. J. Xing, An optimal control problem for mean-field forward-backward stochastic differential equation with noisy observation, Automatica J. IFAC, 86 (2017), 104-109.  doi: 10.1016/j.automatica.2017.07.018.

[39]

G. C. WangH. Xiao and J. Xiong, A kind of LQ non-zero sum differential game of backward stochastic differential equations with asymmetric information, Automatica J. IFAC, 97 (2018), 346-352.  doi: 10.1016/j.automatica.2018.08.019.

[40]

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

[41]

G. C. Wang and Z. Y. 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.

[42] J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford University Press, London, 2008. 
[43]

J. XiongS. Q. Zhang and Y. Zhuang, Partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance, Math. Control Relat. Fields, 9 (2019), 257-276.  doi: 10.3934/mcrf.2019013.

[44]

J. J. Xu, J. T. Shi and H. S. Zhang, A leader-follower stochastic linear quadratic differential game with time delay, Sci. China Infor. Sci., 61 (2018), 112202, 1-13. doi: 10.1007/s11432-017-9293-4.

[45]

J. J. Xu and H. S. Zhang, Sufficient and necessary open-loop Stackelberg strategy for two-player game with time delay, IEEE Trans. Cyber., 46 (2016), 438-449.  doi: 10.1109/TCYB.2015.2403262.

[46]

J. M. Yong, A leader-follower stochastic linear quadratic differential game, SIAM J. Control Optim., 41 (2002), 1015-1041.  doi: 10.1137/S0363012901391925.

[47]

J. M. Yong, Linear forward-backward stochastic differential equations, Appl. Math. Optim., 39 (1999), 93-119.  doi: 10.1007/s002459900100.

[48]

J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

[49]

Z. Y. Yu and S. L. Ji, Linear-quadratic non-zero sum differential game of backward stochstic differential equations. In Proc. 27th Chinese Control Conference, Kunming, China, 2008,562-566.

[50]

J. F. Zhang, Backward Stochastic Differential Equations. From Linear to Fully Nonlinear Theory, Springer, New York, 2017. doi: 10.1007/978-1-4939-7256-2.

[51]

Y. Y. Zheng and J. T. Shi, A Stackelberg game of backward stochastic differential equations with applications, Dyn. Games Appl., online first (2019). doi: 10.1007/s13235-019-00341-z.

[52]

S. S. Zuo and H. Min, Optimal control problems of mean-field forward-backward stochastic differential equations with partial information, in Proc. 25th Chinese Control and Decision Conference, Guiyang, 2013, 5010-5014.

show all references

References:
[1]

A. Aurell, Mean-field type games between two players driven by backward stochastic differential equations, Games, 9 (2018), Paper No. 88, 26 pp. doi: 10.3390/g9040088.

[2]

A. Bagchi and T. Başar, Stackelberg strategies in linear-quadratic stochastic differential games, J. Optim. Theory Appl., 35 (1981), 443-464.  doi: 10.1007/BF00934911.

[3]

A. BensoussanS. K. Chen and S. P. Sethi, The maximum principle for global solutions of stochastic Stackelberg differential games, SIAM J. Control Optim., 53 (2015), 1956-1981.  doi: 10.1137/140958906.

[4]

J. Bismut, An introductory approach to duality in optimal stochastic control, SIAM Rev., 20 (1978), 62-78.  doi: 10.1137/1020004.

[5]

R. BuckdahnB. DjehicheJ. Li and S. G. Peng, Mean-field backward stochastic differential equations: A limit approach, Ann. Probab., 37 (2009), 1524-1565.  doi: 10.1214/08-AOP442.

[6]

S. P. Chen and X. Y. Zhou, Stochastic linear quadratic regulators with indefinite control weight costs. II, SIAM J. Control Optim., 39 (2000), 1065-1081.  doi: 10.1137/S0363012998346578.

[7]

M. Di GiacintoS. Federico and F. Gozzi, Pension funds with a minimum guarantee: A stochastic control approach, Finance Stoch., 15 (2011), 297-342.  doi: 10.1007/s00780-010-0127-7.

[8]

N. G. Dokuchaev and X. Y. Zhou, Stochastic controls with terminal contingent conditions, J. Math. Anal. Appl., 238 (1999), 143-165.  doi: 10.1006/jmaa.1999.6515.

[9]

K. DuJ. H. Huang and Z. Wu, Linear quadratic mean-field-game of backward stochastic differential systems, Math. Control Relat. Fields, 8 (2019), 653-678.  doi: 10.3934/mcrf.2018028.

[10]

K. Du and Z. Wu, Linear-quadratic Stackelberg game for mean-field backward stochastic differential system and application, Math. Probl. Eng., Vol. 2019, Art. Id. 1798585, 17 pp. doi: 10.1155/2019/1798585.

[11]

D. Duffie and L. G. Epstein, Stochastic Differential Utility, Econometrica, 60 (1992), 353-394.  doi: 10.2307/2951600.

[12]

N. El KarouiS. G. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

[13]

S. Hamadene and J. P. Lepeltier, Zero-sum stochastic differential games and backward equations, Systems Control Lett., 24 (1995), 259-263.  doi: 10.1016/0167-6911(94)00011-J.

[14]

J. H. HuangG. C. Wang and J. Xiong, A maximum principle for partial information backward stochastic control problems with applications, SIAM J. Control Optim., 48 (2009), 2106-2117.  doi: 10.1137/080738465.

[15]

J. H. HuangS. J. Wang and Z. Wu, Backward mean-field Linear-Quadratic-Gaussian (LQG) games: Full and partial information, IEEE Trans. Automat. Control, 61 (2016), 3784-3796.  doi: 10.1109/TAC.2016.2519501.

[16]

P. Y. Huang and G. C. Wang, A non-zero sum differential game of mean-field backward stochastic differential equation, in Proc. 2017 Chinese Automation Congress, Jinan, 2017, 4827-4831.

[17]

P. Y. Huang, G. C. Wang and H. J. Zhang, An asymmetric information non-zero sum differential game of mean-field backward stochastic differential equation with applications, Adv. Differ. Equ., 2019, Paper No. 236, 25 pp. doi: 10.1186/s13662-019-2166-5.

[18]

M. Kohlmann and X. Y. Zhou, Relationship between backward stochastic differential equations and stochastic controls: A linear-quadratic approach, SIAM J. Control Optim., 38 (2000), 1392-1407.  doi: 10.1137/S036301299834973X.

[19]

N. Li and Z. Y. Yu, Forward-backward stochastic differential equations and linear-quadratic generalized Stackelberg games, SIAM J. Control Optim., 56 (2018), 4148-4180.  doi: 10.1137/17M1158392.

[20]

X. LiJ. R. Sun and J. Xiong, Linear quadratic optimal control problems for mean-field backward stochastic differential equations, Appl. Math. Optim., 80 (2019), 223-250.  doi: 10.1007/s00245-017-9464-7.

[21]

A. E. B. Lim and X. Y. Zhou, Linear-quadratic control of backward stochastic differential equations, SIAM J. Control Optim., 40 (2001), 450-474.  doi: 10.1137/S0363012900374737.

[22]

Y. N. LinX. S. Jiang and W. H. Zhang, An open-loop Stackelberg strategy for the linear quadratic mean-field stochastic differential game, IEEE Trans. Automat. Control, 64 (2019), 97-110.  doi: 10.1109/TAC.2018.2814959.

[23]

H. P. Ma and B. Liu, Infinite horizon optimal control problem of mean-field backward stochastic delay differential equation under partial information, Eur. J. Control, 36 (2017), 43-50.  doi: 10.1016/j.ejcon.2017.04.001.

[24]

J. Moon and T. Başar, Linear quadratic mean field Stackelberg differential games, Automatica J. IFAC, 97 (2018), 200-213.  doi: 10.1016/j.automatica.2018.08.008.

[25]

B. ØksendalL. Sandal and J. Ubøe, Stochastic Stackelberg equilibria with applications to time dependent newsvendor models, J. Econom. Dynam. Control, 37 (2013), 1284-1299.  doi: 10.1016/j.jedc.2013.02.010.

[26]

E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14 (1990), 55-61.  doi: 10.1016/0167-6911(90)90082-6.

[27]

E. Pardoux and A. Răşcanu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Springer, Cham, 2014. doi: 10.1007/978-3-319-05714-9.

[28]

S. G. Peng, A generalized dynamic programming principle and Hamilton-Jacobi-Bellmen equation, Stochastics Stochastics Rep., 38 (1992), 119-134.  doi: 10.1080/17442509208833749.

[29]

S. G. Peng, Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim., 27 (1993), 125-144.  doi: 10.1007/BF01195978.

[30]

J. T. Shi, Optimal control of backward stochastic differential equations with time delayed generators. In Proc. 30th Chinese Control Conference, Yantai, China, 2011, 1285-1289.

[31]

J. T. Shi, Sufficient conditions of optimality for mean-field stochastic control problems, In Proc. 12th International Conference on Automation, Robots, Control and Vision, Guangzhou, China, 2012,747-752. doi: 10.1109/ICARCV.2012.6485251.

[32]

J. T. Shi and G. C. Wang, A non-zero 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.

[33]

J. T. ShiG. C. Wang and J. Xiong, A Leader-follower stochastic differential game with asymmetric information and applications, Automatica J. IFAC, 63 (2016), 60-73.  doi: 10.1016/j.automatica.2015.10.011.

[34]

J. T. Shi, G. C. Wang and J. Xiong, Linear-quadratic stochastic Stackelberg differential game with asymmetric information, Sci. China Infor. Sci., 60 (2017), 092202, 1-15.

[35]

J. T. Shi, G. C. Wang and J. Xiong, Stochastic linear quadratic Stackelberg differential game with overlapping information, ESAIM: Control, Optim. Calcu. Varia., 26 (2020), Art. No. 83, 38 pp. doi: 10.1051/cocv/2020006.

[36]

M. Simaan and J. B. Cruz Jr., On the Stackelberg game strategy in nonzero-sum games, J. Optim. Theory Appl., 11 (1973), 533-555.  doi: 10.1007/BF00935665.

[37]

H. von Stackelberg, Marktform und Gleichgewicht, Springer, Vienna, 1934.

[38]

G. C. WangH. Xiao and G. J. Xing, An optimal control problem for mean-field forward-backward stochastic differential equation with noisy observation, Automatica J. IFAC, 86 (2017), 104-109.  doi: 10.1016/j.automatica.2017.07.018.

[39]

G. C. WangH. Xiao and J. Xiong, A kind of LQ non-zero sum differential game of backward stochastic differential equations with asymmetric information, Automatica J. IFAC, 97 (2018), 346-352.  doi: 10.1016/j.automatica.2018.08.019.

[40]

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

[41]

G. C. Wang and Z. Y. 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.

[42] J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford University Press, London, 2008. 
[43]

J. XiongS. Q. Zhang and Y. Zhuang, Partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance, Math. Control Relat. Fields, 9 (2019), 257-276.  doi: 10.3934/mcrf.2019013.

[44]

J. J. Xu, J. T. Shi and H. S. Zhang, A leader-follower stochastic linear quadratic differential game with time delay, Sci. China Infor. Sci., 61 (2018), 112202, 1-13. doi: 10.1007/s11432-017-9293-4.

[45]

J. J. Xu and H. S. Zhang, Sufficient and necessary open-loop Stackelberg strategy for two-player game with time delay, IEEE Trans. Cyber., 46 (2016), 438-449.  doi: 10.1109/TCYB.2015.2403262.

[46]

J. M. Yong, A leader-follower stochastic linear quadratic differential game, SIAM J. Control Optim., 41 (2002), 1015-1041.  doi: 10.1137/S0363012901391925.

[47]

J. M. Yong, Linear forward-backward stochastic differential equations, Appl. Math. Optim., 39 (1999), 93-119.  doi: 10.1007/s002459900100.

[48]

J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

[49]

Z. Y. Yu and S. L. Ji, Linear-quadratic non-zero sum differential game of backward stochstic differential equations. In Proc. 27th Chinese Control Conference, Kunming, China, 2008,562-566.

[50]

J. F. Zhang, Backward Stochastic Differential Equations. From Linear to Fully Nonlinear Theory, Springer, New York, 2017. doi: 10.1007/978-1-4939-7256-2.

[51]

Y. Y. Zheng and J. T. Shi, A Stackelberg game of backward stochastic differential equations with applications, Dyn. Games Appl., online first (2019). doi: 10.1007/s13235-019-00341-z.

[52]

S. S. Zuo and H. Min, Optimal control problems of mean-field forward-backward stochastic differential equations with partial information, in Proc. 25th Chinese Control and Decision Conference, Guiyang, 2013, 5010-5014.

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