doi: 10.3934/mcrf.2021026
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Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems

Department of Electrical Engineering, Hanyang University, Seoul 04763, South Korea

Received  July 2020 Revised  October 2020 Early access April 2021

Fund Project: This research was supported in part by the National Research Foundation of Korea (NRF) Grant funded by the Ministry of Science and ICT, South Korea (NRF-2017R1A5A1015311) and in part by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2020-0-01373, Artificial Intelligence Graduate School Program (Hanyang University))

In this paper, we consider linear-quadratic (LQ) leader-follower Stackelberg differential games for mean-field type stochastic systems with jump diffusions, where the system includes mean-field variables, i.e., the expected value of state and control variables. We first solve the LQ mean-field type control problem of the follower using the stochastic maximum principle and obtain the state-feedback representation of the open-loop optimal solution in terms of the coupled integro-Riccati differential equations (CIRDEs) via the Four-Step Scheme. Next, we solve the problem of the leader, which is the LQ control problem subject to the mean-field type forward-backward stochastic system with jump diffusions, where the constraint characterizes the rational behavior of the follower. Using the variational approach, we obtain the (mean-field type) stochastic maximum principle. However, to obtain the state-feedback representation of the open-loop optimal solution of the leader, there is a technical challenge due to the jump process. We consider two different cases, in which the state-feedback type control in terms of the CIRDEs can be characterized by generalizing the Four-Step Scheme. We finally show that the state-feedback type controls of the open-loop optimal solutions for the leader and the follower constitute the Stackelberg equilibrium.

Citation: Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021026
References:
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D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edition, Cambridge University Press, Cambridge, UK, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar

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J. Barreiro-GomezT. E. Duncan and H. Tembine, Linear-quadratic mean-field-type games: Jump-diffusion process with regime switching, IEEE Trans. Automat. Control, 64 (2019), 4329-4336.  doi: 10.1109/TAC.2019.2895295.  Google Scholar

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A. BenssousanB. DjehicheH. Tembine and S. C. P. Yam, Mean-field-type games with jump and regime switching, Dyn. Games Appl., 10 (2020), 19-57.  doi: 10.1007/s13235-019-00306-2.  Google Scholar

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R. BuckdahnB. Djehiche and J. Li, A general stochastic maximum principle for SDEs of mean-field type, Appl. Math. Optim., 64 (2011), 197-216.  doi: 10.1007/s00245-011-9136-y.  Google Scholar

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B. DjehicheH. Tembine and R. Tempone, A stochastic maximum principle for risk-sensitive mean-field type control, IEEE Trans. Automat. Control, 60 (2015), 2640-2649.  doi: 10.1109/TAC.2015.2406973.  Google Scholar

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H. DuJ. Huang and Y. Qin, A stochastic maximum principle for delayed mean-field stochastic differential equations and its applications, IEEE Trans. Automat. Control, 58 (2013), 3212-3217.  doi: 10.1109/TAC.2013.2264550.  Google Scholar

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X. Li, J. Sun and J. Yong, Mean-field stochastic linear quadratic optimal control problems: Closed-loop solvability, Probab. Uncertain. Quant. Risk, 1 (2016), Paper No. 2, 24 pp. doi: 10.1186/s41546-016-0002-3.  Google Scholar

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Q. Meng and Y. Shen, Optimal control of mean-field jump-diffusion systems with delay: A stochastic maximum principle approach, J. Comput. Appl. Math., 279 (2015), 13-30.  doi: 10.1016/j.cam.2014.10.011.  Google Scholar

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J. Moon and Y. Kim, Linear exponential quadratic control for mean field stochastic systems, IEEE Trans. Automat. Control, 64 (2019), 5094-5100.  doi: 10.1109/TAC.2019.2908520.  Google Scholar

[21]

J. Moon and H. J. Yang, Linear-quadratic time-inconsistent mean-field type Stackelberg differential games: Time-consistent open-loop solutions, IEEE Trans. Automat. Control, 66 (2021), 375-382.   Google Scholar

[22]

Y.-H. NiX. Li and J.-F. Zhang, Indefinite mean-field stochastic linear-quadratic optimal control: From finite horizon to infinite horizon, IEEE Trans. Automat. Control, 61 (2016), 3269-3284.  doi: 10.1109/TAC.2015.2509958.  Google Scholar

[23]

Y.-H. NiX. Li and J.-F. Zhang, Meanfield stochastic linear-quadratic optiamal control with Markov jump parameters, Systems Control Lett., 93 (2016), 69-76.  doi: 10.1016/j.sysconle.2016.04.002.  Google Scholar

[24]

Y.-H. NiJ.-F. Zhang and X. Li, Indefinite mean-field stochastic linear-quadratic optimal control, IEEE Trans. Automat. Control, 60 (2015), 1786-1800.  doi: 10.1109/TAC.2014.2385253.  Google Scholar

[25]

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

[26]

N. Privault, Notes on Stochastic Finance, 2021, https://personal.ntu.edu.sg/nprivault/MA5182/stochastic_finance.pdf. Google Scholar

[27]

Y. ShenQ. Meng and P. Shi, Maximum principle for mean-field jump-diffusion stochastic delay differential equations and its application to finance, Automatica J. IFAC, 50 (2014), 1565-1579.  doi: 10.1016/j.automatica.2014.03.021.  Google Scholar

[28]

Y. Shen and T. K. Siu, The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem, Nonlinear Anal., 86 (2013), 58-73.  doi: 10.1016/j.na.2013.02.029.  Google Scholar

[29]

L. YiX. WuX. Li and X. Cui, A mean-field formulation for optimal multi-period mean-variance portfolio selection with an uncertain exit time, Oper. Res. Lett., 42 (2014), 489-494.  doi: 10.1016/j.orl.2014.08.007.  Google Scholar

[30]

J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations, SIAM J. Control Optim., 51 (2013), 2809-2838.  doi: 10.1137/120892477.  Google Scholar

[31]

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

[32]

F. ZhangY. Dong and Q. Meng, Backward stochastic Riccati equation with jumps associated with stochastic linear quadratic optimal control with jumps and random coefficients, SIAM J. Control Optim., 58 (2020), 393-424.  doi: 10.1137/18M1209684.  Google Scholar

[33]

X. ZhangZ. Sun and J. Xiong, A general stochastic maximum principle for a Markov regime switching jump-diffusion model of mean-field type, SIAM J. Control Optim., 56 (2018), 2563-2592.  doi: 10.1137/17M112395X.  Google Scholar

show all references

References:
[1]

D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edition, Cambridge University Press, Cambridge, UK, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar

[2]

J. Barreiro-GomezT. E. Duncan and H. Tembine, Linear-quadratic mean-field-type games: Jump-diffusion process with regime switching, IEEE Trans. Automat. Control, 64 (2019), 4329-4336.  doi: 10.1109/TAC.2019.2895295.  Google Scholar

[3]

T. Bașar and G. J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, SIAM, Philadelphia, PA, 1999. doi: 10.1137/1.9781611971132.  Google Scholar

[4]

T. Bașar and R. Srikant, A Stackelberg network game with a large number of followers, J. Optim. Theory Appl., 115 (2002), 479-490.  doi: 10.1023/A:1021294828483.  Google Scholar

[5]

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

[6]

A. BenssousanB. DjehicheH. Tembine and S. C. P. Yam, Mean-field-type games with jump and regime switching, Dyn. Games Appl., 10 (2020), 19-57.  doi: 10.1007/s13235-019-00306-2.  Google Scholar

[7]

R. BuckdahnB. Djehiche and J. Li, A general stochastic maximum principle for SDEs of mean-field type, Appl. Math. Optim., 64 (2011), 197-216.  doi: 10.1007/s00245-011-9136-y.  Google Scholar

[8]

B. DjehicheH. Tembine and R. Tempone, A stochastic maximum principle for risk-sensitive mean-field type control, IEEE Trans. Automat. Control, 60 (2015), 2640-2649.  doi: 10.1109/TAC.2015.2406973.  Google Scholar

[9]

H. DuJ. Huang and Y. Qin, A stochastic maximum principle for delayed mean-field stochastic differential equations and its applications, IEEE Trans. Automat. Control, 58 (2013), 3212-3217.  doi: 10.1109/TAC.2013.2264550.  Google Scholar

[10]

R. ElliottX. Li and Y. H. Ni, Discrete-time mean field stochastic linear-quadratic optimal control problems, Automatica J. IFAC, 49 (2013), 3222-3233.  doi: 10.1016/j.automatica.2013.08.017.  Google Scholar

[11]

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

[12]

M. HafayedS. Abbas and A. Abba, On mean-field partial information maximum principle of optimal control for stochastic systems with Lévy processes, J. Optim. Theory Appl., 167 (2015), 1051-1069.  doi: 10.1007/s10957-015-0762-4.  Google Scholar

[13]

R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd edition, Cambridge University Press, New York, 2013.  Google Scholar

[14]

J. Li, Stochastic maximum principle in the mean-field controls, Automatica J. IFAC, 48 (2012), 366-373.  doi: 10.1016/j.automatica.2011.11.006.  Google Scholar

[15]

X. Li, J. Sun and J. Yong, Mean-field stochastic linear quadratic optimal control problems: Closed-loop solvability, Probab. Uncertain. Quant. Risk, 1 (2016), Paper No. 2, 24 pp. doi: 10.1186/s41546-016-0002-3.  Google Scholar

[16]

Y. LinX. Jiang and W. 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.  Google Scholar

[17]

Q. Meng and Y. Shen, Optimal control of mean-field jump-diffusion systems with delay: A stochastic maximum principle approach, J. Comput. Appl. Math., 279 (2015), 13-30.  doi: 10.1016/j.cam.2014.10.011.  Google Scholar

[18]

J. Moon, Linear-quadratic mean field stochastic zero-sum differential games, Automatica J. IFAC, 120 (2020), 109067, 10 pp. doi: 10.1016/j.automatica.2020.109067.  Google Scholar

[19]

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

[20]

J. Moon and Y. Kim, Linear exponential quadratic control for mean field stochastic systems, IEEE Trans. Automat. Control, 64 (2019), 5094-5100.  doi: 10.1109/TAC.2019.2908520.  Google Scholar

[21]

J. Moon and H. J. Yang, Linear-quadratic time-inconsistent mean-field type Stackelberg differential games: Time-consistent open-loop solutions, IEEE Trans. Automat. Control, 66 (2021), 375-382.   Google Scholar

[22]

Y.-H. NiX. Li and J.-F. Zhang, Indefinite mean-field stochastic linear-quadratic optimal control: From finite horizon to infinite horizon, IEEE Trans. Automat. Control, 61 (2016), 3269-3284.  doi: 10.1109/TAC.2015.2509958.  Google Scholar

[23]

Y.-H. NiX. Li and J.-F. Zhang, Meanfield stochastic linear-quadratic optiamal control with Markov jump parameters, Systems Control Lett., 93 (2016), 69-76.  doi: 10.1016/j.sysconle.2016.04.002.  Google Scholar

[24]

Y.-H. NiJ.-F. Zhang and X. Li, Indefinite mean-field stochastic linear-quadratic optimal control, IEEE Trans. Automat. Control, 60 (2015), 1786-1800.  doi: 10.1109/TAC.2014.2385253.  Google Scholar

[25]

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

[26]

N. Privault, Notes on Stochastic Finance, 2021, https://personal.ntu.edu.sg/nprivault/MA5182/stochastic_finance.pdf. Google Scholar

[27]

Y. ShenQ. Meng and P. Shi, Maximum principle for mean-field jump-diffusion stochastic delay differential equations and its application to finance, Automatica J. IFAC, 50 (2014), 1565-1579.  doi: 10.1016/j.automatica.2014.03.021.  Google Scholar

[28]

Y. Shen and T. K. Siu, The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem, Nonlinear Anal., 86 (2013), 58-73.  doi: 10.1016/j.na.2013.02.029.  Google Scholar

[29]

L. YiX. WuX. Li and X. Cui, A mean-field formulation for optimal multi-period mean-variance portfolio selection with an uncertain exit time, Oper. Res. Lett., 42 (2014), 489-494.  doi: 10.1016/j.orl.2014.08.007.  Google Scholar

[30]

J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations, SIAM J. Control Optim., 51 (2013), 2809-2838.  doi: 10.1137/120892477.  Google Scholar

[31]

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

[32]

F. ZhangY. Dong and Q. Meng, Backward stochastic Riccati equation with jumps associated with stochastic linear quadratic optimal control with jumps and random coefficients, SIAM J. Control Optim., 58 (2020), 393-424.  doi: 10.1137/18M1209684.  Google Scholar

[33]

X. ZhangZ. Sun and J. Xiong, A general stochastic maximum principle for a Markov regime switching jump-diffusion model of mean-field type, SIAM J. Control Optim., 56 (2018), 2563-2592.  doi: 10.1137/17M112395X.  Google Scholar

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