Open-loop equilibrium strategy of time-inconsistent deterministic linear-quadratic problems

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Abstract

This paper investigates time-inconsistent linear-quadratic control problems within an open-loop framework, where the objective functional incorporates a non-exponential discounting structure. By introducing a two-point boundary value problem and a Riccati-type equation with an error function, it is established, under appropriate conditions, that the existence of an open-loop equilibrium strategy for the time-inconsistent control problem is equivalent to the existence of solutions to both the two-point boundary value problem and the Riccati-type equation. Furthermore, through an illustrative example, it is demonstrated that there exists an essential distinction between open-loop and closed-loop equilibria in time-inconsistent control problems.

Keywords

time-inconsistent / open-loop equilibrium / linear-quadratic / error function / two-point boundary value problem / Riccati-type equation

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References

[1] R. H. Strotz, Myopia and inconsistency in dynamic utility maximization, Rev. Econ. Stud., 23 (1955), 165-180. https://doi.org/10.2307/2295722

[2] W. Ji, On the equilibrium strategy of linear-quadratic time-inconsistent control problems, AIMS Math., 10 (2025), 5480-5494. https://doi.org/10.3934/math.2025253

[3] W. Ji, Optimal control problems with time inconsistency, Electron Res. Arch., 31 (2023), 492-508. https://doi.org/10.3934/era.2023024

[4] Q. Zhu, Event-triggered sampling problem for exponential stability of stochastic nonlinear delay systems driven by Lévy processes, IEEE Trans. Autom. Control, 70 (2025), 1176-1183. https://doi.org/10.1109/TAC.2024.3448128

[5] 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

[6] T. Björk, A, Murgoci, A general theory of Markovian time inconsistent stochastic control in discrete time, Finance Stoch., 18 (2014), 545-592. https://doi.org/10.1007/s00780-014-0234-y

[7] J. Yong, Time-inconsistent optimal control problem and the equilibrium HJB equation, Math. Control Relat. Fields, 2 (2012), 271-329. https://doi.org/10.3934/mcrf.2012.2.271.MR3072755

[8] J. Yong, Linear-quadritic optimal control problems for mean-field stochastic differential equations-time consistent solutions, Trans. Am. Math. Soc., 369 (2017), 5467-5523. https://doi.org/10.1051/jp4:20011110

[9] F. Dou, Q. Lü, Time-inconsistent linear quadratic optimal control problems for stochastic evolution equations, SIAM J. Control Optim., 58 (2020), 485-509. https://doi.org/10.1137/19M1250339

[10] I. Ekeland, A. Lazrak, Being serious about non-commitment: Subgame perfect equilibrium in continuous time, preprint, arXiv:math/0604264. https://doi.org/10.48550/arXiv.math/0604264

[11] T. Björk, M. Khapko, A. Murgoci, On time inconsistent stochastic control in continuous time, Finance Stoch., 21 (2017), 331-360. https://doi.org/10.1007/s00780-017-0327-5

[12] H. Cai, D. Chen, Y. Peng, W. Wei, On the time-inconsistent deterministic linear-quadratic control, SIAM J. Control Optim., 60 (2022), 968-991. https://doi.org/10.1137/21M1419611

[13] Y. Hu, H. Jin, X. Y. Zhou, Time inconsistent stochastic linear quadratic control, SIAM J. Control Optim., 50 (2012), 1548-1572. https://doi.org/10.1137/110853960

[14] Y. Hu, H. Jin, X. Zhou, Time-inconsistent stochastic linear-quadratic control: characterization and uniqueness of equilibrium, SIAM J. Control Optim., 55 (2017), 1261-1279. https://doi.org/10.1137/15M1019040

[15] X. He, Z. Jiang, On the equilibrium strategies for time-inconsistent problems in continuous time, SIAM J. Control Optim., 59 (2021), 3860-3886. https://doi.org/10.1137/20M1382106

[16] E. Bayraktar, J. Zhang, Z. Zhou, Equilibrium concepts for time-inconsistent stopping problems in continuous time, Math. Finance, 31 (2021), 69-95. https://doi.org/10.1111/mafi.12293

[17] Z. Zhou, Almost strong equilibria for time-inconsistent stopping problems under finite horizon in continuous time, Math. Finance, 34 (2024), 1089-1122. https://doi.org/10.1111/mafi.12428

[18] Y. Ni, X. Li, J. Zhang, M. Krstic, Mixed equilibrium solution of time-inconsistent stochastic linear-quadratic problem, SIAM J. Control Optim., 57 (2019), 533-569. https://doi.org/10.1137/18M1177068

[19] Y. Peng, W. Wei, Solutions to equilibrium HJB equations for time-inconsistent deterministic linear quadratic control: characterization and uniqueness, preprint, arXiv:2308.13850. https://doi.org/10.48550/arXiv.2308.13850