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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[21]

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

[22]

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

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

[30]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[21]

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

[22]

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

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

[30]

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

[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.

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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