Linear-quadratic-Gaussian mean-field games driven by Poisson jumps with major and minor agents

Ruimin Xu; Kaiyue Dong; Jingyu Zhang; Ying Zhou; Ruimin Xu; Kaiyue Dong; Jingyu Zhang; Ying Zhou

doi:10.3934/math.2025503

AIMS Mathematics

2025, Volume 10, Issue 5: 11086-11110. doi: 10.3934/math.2025503

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Linear-quadratic-Gaussian mean-field games driven by Poisson jumps with major and minor agents

School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China

Received: 22 January 2025 Revised: 22 April 2025 Accepted: 27 April 2025 Published: 15 May 2025
MSC : 91A16

This paper studies mean-field linear-quadratic-Gaussian (LQG) games with a major agent and a large number of minor agents, where each agent's state process is driven by a Poisson random measure and independent Brownian motion. The major and minor agents were coupled via both their state dynamics as well as in their individual cost functionals. By the Nash certainty equivalence (NCE) methodology, two limiting control problems were constructed and the decentralized strategies were derived through the consistency condition. The $ \epsilon $-Nash equilibrium property of the obtained decentralized strategies was shown for a finite $ N $ population system where $ \epsilon = O(1/\sqrt{N}) $. A numerical example was presented to illustrate the consistency of the mean-field estimation and the impact of the population's collective behavior.
- mean-field games,
- major and minor agents,
- decentralized control,
- jump-diffusion,
- $ \epsilon $-Nash equilibrium
Citation: Ruimin Xu, Kaiyue Dong, Jingyu Zhang, Ying Zhou. Linear-quadratic-Gaussian mean-field games driven by Poisson jumps with major and minor agents[J]. AIMS Mathematics, 2025, 10(5): 11086-11110. doi: 10.3934/math.2025503

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Abstract

This paper studies mean-field linear-quadratic-Gaussian (LQG) games with a major agent and a large number of minor agents, where each agent's state process is driven by a Poisson random measure and independent Brownian motion. The major and minor agents were coupled via both their state dynamics as well as in their individual cost functionals. By the Nash certainty equivalence (NCE) methodology, two limiting control problems were constructed and the decentralized strategies were derived through the consistency condition. The $ \epsilon $-Nash equilibrium property of the obtained decentralized strategies was shown for a finite $ N $ population system where $ \epsilon = O(1/\sqrt{N}) $. A numerical example was presented to illustrate the consistency of the mean-field estimation and the impact of the population's collective behavior.

References

[1]	C. Benazzoli, L. Campi, L. Di Persio, $\epsilon$-Nash equilibrium in stochastic differential games with mean-field interaction and controlled jumps, Stat. Probabil. Lett., 154 (2019), 108522. https://doi.org/10.1016/j.spl.2019.05.021 doi: 10.1016/j.spl.2019.05.021
[2]	R. Carmona, F. Delarue, Probabilistic analysis of mean-field games, SIAM J. Control Optim., 51 (2013), 2705–2734. https://doi.org/10.1137/120883499 doi: 10.1137/120883499
[3]	Y. Hu, J. Huang, T. Nie, Linear-quadratic-gaussian mixed mean-field games with heterogeneous input constraints, SIAM J. Control Optim., 56 (2018), 2835–2877. https://doi.org/10.1137/17M1151420 doi: 10.1137/17M1151420
[4]	M. Huang, P. E. Caines, R. P. Malhame, Social optima in mean field LQG control: centralized and decentralized strategies, IEEE T. Automat. Contr., 57 (2012), 1736–1751. https://doi.org/10.1109/TAC.2012.2183439 doi: 10.1109/TAC.2012.2183439
[5]	J. Huang, Z. Qiu, S. Wang, Z. Wu, A unified relation analysis of linear-quadratic mean-field game, team, and control, IEEE T. Automat. Contr., 69 (2024), 3325–3332. https://doi.org/10.1109/TAC.2023.3323576 doi: 10.1109/TAC.2023.3323576
[6]	B. Wang, J. Zhang, Social optima in mean field linear-quadratic-gaussian models with Markov jump parameters, SIAM J. Control Optim., 55 (2017), 429–456. https://doi.org/10.1137/15M104178X doi: 10.1137/15M104178X
[7]	H. Wang, R. Xu, Time-inconsistent LQ games for large-population systems and applications, J. Optim. Theory Appl., 197 (2023), 1249–1268. https://doi.org/10.1007/s10957-023-02223-2 doi: 10.1007/s10957-023-02223-2
[8]	R. Xu, F. Zhang, $\epsilon$-Nash mean-field games for general linear-quadratic systems with applications, Automatica, 114 (2020), 108835. https://doi.org/10.1016/j.automatica.2020.108835 doi: 10.1016/j.automatica.2020.108835
[9]	H. Yuan, Q. Zhu, The well-posedness and stabilities of mean-field stochastic differential equations driven by g-Brownian motion, SIAM J. Control Optim., 63 (2025), 596–624. https://doi.org/10.1137/23M1593681 doi: 10.1137/23M1593681
[10]	H. Yuan, Q. Zhu, The stabilities of delay stochastic McKean-Vlasov equations in the g-framework, Sci. China Inf. Sci., 68 (2025), 112203. https://doi.org/10.1007/s11432-024-4075-2 doi: 10.1007/s11432-024-4075-2
[11]	M. Huang, P. E. Caines, R. P. Malhame, Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized $\epsilon$-Nash equilibria, IEEE T. Automat. Contr., 52 (2007), 1560–1571. https://doi.org/10.1109/TAC.2007.904450 doi: 10.1109/TAC.2007.904450
[12]	M. Huang, Large-population LQG games involving a major player: the Nash certainty equivalence principle, SIAM J. Control Optim., 48 (2010), 3318–3353. https://doi.org/10.1137/080735370 doi: 10.1137/080735370
[13]	M. Nourian, P. E. Caines, $\epsilon$-Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents, SIAM J. Control Optim., 51 (2013), 3302–3331. https://doi.org/10.1137/120889496 doi: 10.1137/120889496
[14]	J. M. Lasry, P. L. Lions, Jeux à Champ Moyen. Ⅰ–Le cas stationnaire, C. R. Math., 343 (2006), 619–625. https://doi.org/10.1016/j.crma.2006.09.019 doi: 10.1016/j.crma.2006.09.019
[15]	J. M. Lasry, P. L. Lions, Jeux à Champ Moyen. Ⅱ–Horizon fini et contrôle optimal, C. R. Math., 343 (2006), 679–684. https://doi.org/10.1016/j.crma.2006.09.018 doi: 10.1016/j.crma.2006.09.018
[16]	J. M. Lasry, P. L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229–260. https://doi.org/10.1007/s11537-007-0657-8 doi: 10.1007/s11537-007-0657-8
[17]	A. Bensoussan, J. Frehse, P. Yam, Mean field games and mean field type control theory, New York: Springer, 2013. https://doi.org/10.1007/978-1-4614-8508-7
[18]	J. Huang, S. Wang, Z. Wu, Backward mean-field linear-quadratic-gaussian (LQG) games: full and partial information, IEEE T. Automat. Contr., 61 (2016), 3784–3796. https://doi.org/10.1109/TAC.2016.2519501 doi: 10.1109/TAC.2016.2519501
[19]	T. Nie, S. Wang, Z. Wu, Linear-quadratic delayed mean-field social optimization, Appl. Math. Optim., 89 (2024), 4. https://doi.org/10.1007/s00245-023-10067-5 doi: 10.1007/s00245-023-10067-5
[20]	R. Xu, J. Shi, $\epsilon$-Nash mean-field games for linear-quadratic systems with random jumps and applications, Int. J. Control, 94 (2021), 1415–1425. https://doi.org/10.1080/00207179.2019.1651940 doi: 10.1080/00207179.2019.1651940
[21]	R. Xu, T. Wu, Risk-sensitive large-population linear-quadratic-gaussian games with major and minor agents, Asian J. Control, 25 (2023), 4391–4403. https://doi.org/10.1002/asjc.3106 doi: 10.1002/asjc.3106
[22]	H. Wang, R. Xu, Time-inconsistent large-population linear-quadratic games with major and minor agents, Int. J. Control, 2025. https://doi.org/10.1080/00207179.2025.2491823 doi: 10.1080/00207179.2025.2491823
[23]	B. Düring, P. Markowich, J. F. Pietschmann, M. T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, P. Roy. Soc. A Math. Phy., 465 (2009), 3687–3708. https://doi.org/10.1098/rspa.2009.0239 doi: 10.1098/rspa.2009.0239
[24]	Z. Ma, D. S. Callaway, I. A. Hiskens, Decentralized charging control of large populations of plug-in electric vehicles, IEEE T. Control Syst. Technol., 21 (2011), 67–78. https://doi.org/10.1109/TCST.2011.2174059 doi: 10.1109/TCST.2011.2174059
[25]	J. Shi, Z. Wu, Maximum principle for forward-backward stochastic control system with random jumps and applications to finance, J. Syst. Sci. Complex., 23 (2010), 219–231. https://doi.org/10.1007/s11424-010-7224-8 doi: 10.1007/s11424-010-7224-8
[26]	J. Shi, Z. Wu, A risk-sensitive stochastic maximum principle for optimal control of jump diffusions and its applications, Acta Math. Sci., 31 (2011), 419–433. https://doi.org/10.1016/S0252-9602(11)60242-7 doi: 10.1016/S0252-9602(11)60242-7
[27]	R. Cont, P. Tankov, Financial modelling with jump processes, Chapman and Hall/CRC, 2003.
[28]	W. He, Y. Wang, Distributed optimal variational GNE seeking in merely monotone games, IEEE/CAA J. Automatic., 11 (2024), 1621–1630. https://doi.org/10.1109/JAS.2024.124284 doi: 10.1109/JAS.2024.124284
[29]	R. Merton, Option pricing when underlying stock returns are discontinuous, J. Financ. Econ., 3 (1976), 125–144. https://doi.org/10.1016/0304-405X(76)90022-2 doi: 10.1016/0304-405X(76)90022-2
[30]	E. Platen, N. Bruti-Liberati, Numerical solution of stochastic differential equations with jumps in finance, Berlin: Springer-Verlag, 2010. https://doi.org/10.1007/978-3-642-13694-8

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