doi: 10.3934/mcrf.2021035
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A linear quadratic stochastic Stackelberg differential game with time delay

School of Mathematics, Shandong University, Jinan 250100, China

* Corresponding author: Jingtao Shi

Received  January 2021 Revised  March 2021 Early access June 2021

Fund Project: This work is supported by National Key R & D Program of China (Grant No. 2018YFB1305400), National Natural Science Foundations of China (Grant Nos. 11971266, 11831010, 11571205), and Shandong Provincial Natural Science Foundations (Grant Nos. ZR2020ZD24, ZR2019ZD42)

This paper is concerned with a linear quadratic stochastic Stackelberg differential game with time delay. The model is general, in which the state delay and the control delay both appear in the state equation, moreover, they both enter into the diffusion term. By introducing two Pseudo-Riccati equations and a special matrix equation, the state feedback representation of the open-loop Stackelberg strategy is derived, under some assumptions. Finally, two examples are given to illustrate the applications of the theoretical results.

Citation: Weijun Meng, Jingtao Shi. A linear quadratic stochastic Stackelberg differential game with time delay. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021035
References:
[1]

Y. Bai, Z. Zhou, H. Xiao and R. Gao, A Stackelberg reinsurance-investment game with asymmetric information and delay, Optimization, (2020). doi: 10.1080/02331934.2020.1777125.  Google Scholar

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T. Bașar, Stochastic stagewise Stackelberg strategies for linear quadratic systems, in Stochastic Control Theory and Stochastic Differential Systems, Lecture Notes in Control and Information Sci., 16, Springer, Berlin-New York, 1979,264–276. doi: 10.1007/BFb0009386.  Google Scholar

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T. Bașar and G. J. Olsder, Dynamic Noncooperative Game Theory, Classics in Applied Mathematics, 23, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. doi: 10.1137/1.9781611971132.  Google Scholar

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A. BensoussanM. H. M. ChauY. Lai and S. C. P. Yam, Linear-quadratic mean field Stackelberg games with state and control delays, SIAM J. Control Optim., 55 (2017), 2748-2781.  doi: 10.1137/15M1052937.  Google Scholar

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A. BensoussanM. H. M. Chau and S. C. P. Yam, Mean field Stackelberg games: Aggregation of delayed instructions, SIAM J. Control Optim., 53 (2015), 2237-2266.  doi: 10.1137/140993399.  Google Scholar

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A. BensoussanS. 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.  Google Scholar

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D. Castanon and M. Athans, On stochastic dynamic Stackelberg strategies, Automatica J. IFAC, 12 (1976), 177-183.  doi: 10.1016/0005-1098(76)90081-9.  Google Scholar

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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

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G. FreilingG. Jank and S. R. Lee, Existence and uniqueness of open-loop Stackelberg equilibria in linear-quadratic differential games, J. Optim. Theory Appl., 110 (2001), 515-544.  doi: 10.1023/A:1017532210579.  Google Scholar

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J. Harband, The existence of monotonic solutions of a nonlinear car-follwing equation, J. Math. Anal. Appl., 57 (1977), 257-272.  doi: 10.1016/0022-247X(77)90259-1.  Google Scholar

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J. HuangX. Li and J. Shi, Forward-backward linear quadratic stochastic optimal control problem with delay, Systems Control Lett., 61 (2012), 623-630.  doi: 10.1016/j.sysconle.2012.02.010.  Google Scholar

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T. Ishida and E. Shimemura, Open-loop Stackelberg strategies in a linear-quadratic differential game with time delay, Internat. J. Control, 45 (1987), 1847-1855.  doi: 10.1080/00207178708933850.  Google Scholar

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T. Ishida and E. Shimemura, Sufficient conditons for the team-optimal closed-loop Stackelberg strategies in linear differential games with time-delay, Internat. J. Control, 37 (1983), 441-454.  doi: 10.1080/00207178308932984.  Google Scholar

[14]

X. LiW. WangJ. Xu and H. Zhang, A Stackelberg strategy for continuous-time mixed $H_2/H_\infty$ control problem with time delay, Control Theory Technol., 16 (2018), 191-202.  doi: 10.1007/s11768-018-8014-4.  Google Scholar

[15]

S. E. A. Mohammed, Stochastic differential equations with memory: Theory, examples and applications, in Stochastic Analysis and Related Topics, Progr. Probab., 42, Birkhäuser Boston, Boston, MA, 1998, 1–77. doi: 10.1007/978-1-4612-2022-0_1.  Google Scholar

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S. E. A. Mohammed, Stochastic Functional Differential Equations, Research Notes in Mathematics, 99, Pitman (Advanced Publishing Program), Boston, MA, 1984.  Google Scholar

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B. ØksendalL. Sandal and J. Ubøe, Stackelberg equilibria in a continuous-time vertical contracting model with uncertain demand and delayed information, J. App. Probab., 51A (2014), 213-226.  doi: 10.1239/jap/1417528477.  Google Scholar

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B. Øksendal and A. Sulem, A maximum principle for optimal control of stochastic systems with delay, with applications to finance, in Optimal Control and Partial Differential Equations, ISO, Amsterdam, 2000, 64–79.  Google Scholar

[19]

L. P. Pan and J. M. Yong, A differential game with multi-level of hierarchy, J. Math. Anal. Appl., 161 (1991), 522-544.  doi: 10.1016/0022-247X(91)90348-4.  Google Scholar

[20]

G. P. Papavassilopoulos and J. B. Cruz Jr., Nonclassical control problems and Stackelberg games, IEEE Trans. Automat. Control, 24 (1979), 155-166.  doi: 10.1109/TAC.1979.1101986.  Google Scholar

[21]

S. G. Peng and Z. Yang, Anticipated backward stochastic differential equations, Ann. Probab., 37 (2009), 877-902.  doi: 10.1214/08-AOP423.  Google Scholar

[22]

F. M. Scherer, Research and development resource allocation under rivalry, Quart. J. Econ., 81 (1967), 359-394.  doi: 10.2307/1884807.  Google Scholar

[23]

J. ShiG. Wang and J. Xiong, 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.  Google Scholar

[24]

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

[25]

J. Shi, G. Wang and J. Xiong, Stochastic linear quadratic Stackelberg differential game with overlapping information, ESAIM Control Optim. Calc. Var., 26 (2020), 38pp. doi: 10.1051/cocv/2020006.  Google Scholar

[26]

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

[27]

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

[28]

J. Xu, J. Shi and H. Zhang, A leader-follower stochastic linear quadratic differential game with time delay, Sci. China Inf. Sci., 61 (2018), 13pp. doi: 10.1007/s11432-017-9293-4.  Google Scholar

[29]

J. Xu and H. 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.  Google Scholar

[30]

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

[31]

H. Zhang and J. Xu, Control for Itô stochastic systems with input delay, IEEE Trans. Automat. Control, 62 (2017), 350-365.  doi: 10.1109/TAC.2016.2551371.  Google Scholar

show all references

References:
[1]

Y. Bai, Z. Zhou, H. Xiao and R. Gao, A Stackelberg reinsurance-investment game with asymmetric information and delay, Optimization, (2020). doi: 10.1080/02331934.2020.1777125.  Google Scholar

[2]

T. Bașar, Stochastic stagewise Stackelberg strategies for linear quadratic systems, in Stochastic Control Theory and Stochastic Differential Systems, Lecture Notes in Control and Information Sci., 16, Springer, Berlin-New York, 1979,264–276. doi: 10.1007/BFb0009386.  Google Scholar

[3]

T. Bașar and G. J. Olsder, Dynamic Noncooperative Game Theory, Classics in Applied Mathematics, 23, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. doi: 10.1137/1.9781611971132.  Google Scholar

[4]

A. BensoussanM. H. M. ChauY. Lai and S. C. P. Yam, Linear-quadratic mean field Stackelberg games with state and control delays, SIAM J. Control Optim., 55 (2017), 2748-2781.  doi: 10.1137/15M1052937.  Google Scholar

[5]

A. BensoussanM. H. M. Chau and S. C. P. Yam, Mean field Stackelberg games: Aggregation of delayed instructions, SIAM J. Control Optim., 53 (2015), 2237-2266.  doi: 10.1137/140993399.  Google Scholar

[6]

A. BensoussanS. 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.  Google Scholar

[7]

D. Castanon and M. Athans, On stochastic dynamic Stackelberg strategies, Automatica J. IFAC, 12 (1976), 177-183.  doi: 10.1016/0005-1098(76)90081-9.  Google Scholar

[8]

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

[9]

G. FreilingG. Jank and S. R. Lee, Existence and uniqueness of open-loop Stackelberg equilibria in linear-quadratic differential games, J. Optim. Theory Appl., 110 (2001), 515-544.  doi: 10.1023/A:1017532210579.  Google Scholar

[10]

J. Harband, The existence of monotonic solutions of a nonlinear car-follwing equation, J. Math. Anal. Appl., 57 (1977), 257-272.  doi: 10.1016/0022-247X(77)90259-1.  Google Scholar

[11]

J. HuangX. Li and J. Shi, Forward-backward linear quadratic stochastic optimal control problem with delay, Systems Control Lett., 61 (2012), 623-630.  doi: 10.1016/j.sysconle.2012.02.010.  Google Scholar

[12]

T. Ishida and E. Shimemura, Open-loop Stackelberg strategies in a linear-quadratic differential game with time delay, Internat. J. Control, 45 (1987), 1847-1855.  doi: 10.1080/00207178708933850.  Google Scholar

[13]

T. Ishida and E. Shimemura, Sufficient conditons for the team-optimal closed-loop Stackelberg strategies in linear differential games with time-delay, Internat. J. Control, 37 (1983), 441-454.  doi: 10.1080/00207178308932984.  Google Scholar

[14]

X. LiW. WangJ. Xu and H. Zhang, A Stackelberg strategy for continuous-time mixed $H_2/H_\infty$ control problem with time delay, Control Theory Technol., 16 (2018), 191-202.  doi: 10.1007/s11768-018-8014-4.  Google Scholar

[15]

S. E. A. Mohammed, Stochastic differential equations with memory: Theory, examples and applications, in Stochastic Analysis and Related Topics, Progr. Probab., 42, Birkhäuser Boston, Boston, MA, 1998, 1–77. doi: 10.1007/978-1-4612-2022-0_1.  Google Scholar

[16]

S. E. A. Mohammed, Stochastic Functional Differential Equations, Research Notes in Mathematics, 99, Pitman (Advanced Publishing Program), Boston, MA, 1984.  Google Scholar

[17]

B. ØksendalL. Sandal and J. Ubøe, Stackelberg equilibria in a continuous-time vertical contracting model with uncertain demand and delayed information, J. App. Probab., 51A (2014), 213-226.  doi: 10.1239/jap/1417528477.  Google Scholar

[18]

B. Øksendal and A. Sulem, A maximum principle for optimal control of stochastic systems with delay, with applications to finance, in Optimal Control and Partial Differential Equations, ISO, Amsterdam, 2000, 64–79.  Google Scholar

[19]

L. P. Pan and J. M. Yong, A differential game with multi-level of hierarchy, J. Math. Anal. Appl., 161 (1991), 522-544.  doi: 10.1016/0022-247X(91)90348-4.  Google Scholar

[20]

G. P. Papavassilopoulos and J. B. Cruz Jr., Nonclassical control problems and Stackelberg games, IEEE Trans. Automat. Control, 24 (1979), 155-166.  doi: 10.1109/TAC.1979.1101986.  Google Scholar

[21]

S. G. Peng and Z. Yang, Anticipated backward stochastic differential equations, Ann. Probab., 37 (2009), 877-902.  doi: 10.1214/08-AOP423.  Google Scholar

[22]

F. M. Scherer, Research and development resource allocation under rivalry, Quart. J. Econ., 81 (1967), 359-394.  doi: 10.2307/1884807.  Google Scholar

[23]

J. ShiG. Wang and J. Xiong, 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.  Google Scholar

[24]

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

[25]

J. Shi, G. Wang and J. Xiong, Stochastic linear quadratic Stackelberg differential game with overlapping information, ESAIM Control Optim. Calc. Var., 26 (2020), 38pp. doi: 10.1051/cocv/2020006.  Google Scholar

[26]

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

[27]

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

[28]

J. Xu, J. Shi and H. Zhang, A leader-follower stochastic linear quadratic differential game with time delay, Sci. China Inf. Sci., 61 (2018), 13pp. doi: 10.1007/s11432-017-9293-4.  Google Scholar

[29]

J. Xu and H. 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.  Google Scholar

[30]

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

[31]

H. Zhang and J. Xu, Control for Itô stochastic systems with input delay, IEEE Trans. Automat. Control, 62 (2017), 350-365.  doi: 10.1109/TAC.2016.2551371.  Google Scholar

Figure 1.  Algorithm scheme of the solution to (53)
Figure 2.  The solutions to (51) and (53)
Figure 3.  The solution to (53)
Figure 4.  The optimal strategy
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