June  2019, 9(2): 385-409. doi: 10.3934/mcrf.2019018

Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I

School of Mathematics, Sichuan University, Chengdu, China

Received  November 2017 Revised  August 2018 Published  December 2018

Fund Project: The research was supported by the NSF of China under grant 11231007, 11401404 and 11471231, and the Fundamental Research Funds for the central Universities (YJ201605)

In this paper, a class of time inconsistent linear quadratic optimal control problems for mean-field stochastic differential equations (SDEs) are considered under Markovian framework. Open-loop equilibrium controls and their particular closed-loop representations are introduced and characterized via variational ideas. Several interesting features are revealed and a system of coupled Riccati equations is derived. In contrast with the analogue optimal control problems of SDEs, the mean-field terms in state equation, which is another reason of time inconsistency, prompts us to define the above two notions in new manners. An interesting result, which is almost trivial in the counterpart problems of SDEs, is given and plays significant role in the previous characterizations. As application, the uniqueness of open-loop equilibrium controls is discussed.

Citation: Tianxiao Wang. Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I. Mathematical Control & Related Fields, 2019, 9 (2) : 385-409. doi: 10.3934/mcrf.2019018
References:
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I. AliaF. Chighoub and A. Sohail, A characterization of equilibrium strategies in continuous-time mean-variance problems for insurers, Insurance Math. Econom., 68 (2016), 212-223. doi: 10.1016/j.insmatheco.2016.03.009.

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T. BjörkM. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance Stoch., 21 (2017), 331-360. doi: 10.1007/s00780-017-0327-5.

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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.

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B. Djehiche and M. Huang, A characterization of sub-game perfect equilibria for SDEs of mean-field type, Dyn. Games Appl., 6 (2016), 55-81. doi: 10.1007/s13235-015-0140-8.

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I. EkelandO. Mbodji and T. Pirvu, Time-consistent portfolio management, SIAM J. Financial Math., 3 (2012), 1-32. doi: 10.1137/100810034.

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I. Ekeland and T. Pirvu, Investment and consumption without commitment, Math. Finance Econ., 2 (2008), 57-86. doi: 10.1007/s11579-008-0014-6.

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Y. HuH. Jin and X. Zhou, Time-inconsistent stochastic linear-quadratic control, SIAM J. Control Optim., 50 (2012), 1548-1572. doi: 10.1137/110853960.

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Y. HuH. Jin and X. Zhou, Time-inconsistent stochastic linear-quadratic control: Characterization and uniqueness of equilibrium, SIAM J. Control Optim., 55 (2017), 1261-1279. doi: 10.1137/15M1019040.

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J. HuangX. Li and T. Wang, Characterizations of closed-loop equilibrium solutions for dynamic mean-variance optimization problems, Systems Control Lett., 110 (2017), 15-20. doi: 10.1016/j.sysconle.2017.09.008.

[10]

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

[11]

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.

[12]

H. Wang and Z. Wu, Partially observed time-inconsistency recursive optimization problem and application, J Optim. Theory Appl., 161 (2014), 664-687. doi: 10.1007/s10957-013-0326-4.

[13]

H. Wang and Z. Wu, Time-inconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation, Math. Control Relat. Fields., 5 (2015), 651-678. doi: 10.3934/mcrf.2015.5.651.

[14]

T. Wang, Equilibrium controls in time inconsistent stochastic linear quadratic problems, Appl. Math. Optim., (2018), 1-29. doi: 10.1007/s00245-018-9513-x.

[15]

J. Wei and T. Wang, Time-consistent mean-variance asset-liability management with random coefficients, Insurance Math. Econom., 77 (2017), 84-96. doi: 10.1016/j.insmatheco.2017.08.011.

[16]

J. Yong, A deterministic linear quadratic time-inconsistent optimal control problem, Math. Control Related Fields, 1 (2011), 83-118. doi: 10.3934/mcrf.2011.1.83.

[17]

J. Yong, Deterministic time-inconsistent optimal control problems--an essentially cooperative approach, Acta Math. Appl. Sin. Engl. Ser., 28 (2012), 1-30. doi: 10.1007/s10255-012-0120-3.

[18]

J. Yong, Time-inconsistent optimal control problem and the equilibrium HJB equation, Math. Control Related Fields, 2 (2012), 271-329. doi: 10.3934/mcrf.2012.2.271.

[19]

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.

[20]

J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations--time-consistent solutions, Trans. Amer. Math. Soc., 369 (2017), 5467-5523. doi: 10.1090/tran/6502.

show all references

References:
[1]

I. AliaF. Chighoub and A. Sohail, A characterization of equilibrium strategies in continuous-time mean-variance problems for insurers, Insurance Math. Econom., 68 (2016), 212-223. doi: 10.1016/j.insmatheco.2016.03.009.

[2]

T. BjörkM. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance Stoch., 21 (2017), 331-360. doi: 10.1007/s00780-017-0327-5.

[3]

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.

[4]

B. Djehiche and M. Huang, A characterization of sub-game perfect equilibria for SDEs of mean-field type, Dyn. Games Appl., 6 (2016), 55-81. doi: 10.1007/s13235-015-0140-8.

[5]

I. EkelandO. Mbodji and T. Pirvu, Time-consistent portfolio management, SIAM J. Financial Math., 3 (2012), 1-32. doi: 10.1137/100810034.

[6]

I. Ekeland and T. Pirvu, Investment and consumption without commitment, Math. Finance Econ., 2 (2008), 57-86. doi: 10.1007/s11579-008-0014-6.

[7]

Y. HuH. Jin and X. Zhou, Time-inconsistent stochastic linear-quadratic control, SIAM J. Control Optim., 50 (2012), 1548-1572. doi: 10.1137/110853960.

[8]

Y. HuH. Jin and X. Zhou, Time-inconsistent stochastic linear-quadratic control: Characterization and uniqueness of equilibrium, SIAM J. Control Optim., 55 (2017), 1261-1279. doi: 10.1137/15M1019040.

[9]

J. HuangX. Li and T. Wang, Characterizations of closed-loop equilibrium solutions for dynamic mean-variance optimization problems, Systems Control Lett., 110 (2017), 15-20. doi: 10.1016/j.sysconle.2017.09.008.

[10]

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

[11]

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.

[12]

H. Wang and Z. Wu, Partially observed time-inconsistency recursive optimization problem and application, J Optim. Theory Appl., 161 (2014), 664-687. doi: 10.1007/s10957-013-0326-4.

[13]

H. Wang and Z. Wu, Time-inconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation, Math. Control Relat. Fields., 5 (2015), 651-678. doi: 10.3934/mcrf.2015.5.651.

[14]

T. Wang, Equilibrium controls in time inconsistent stochastic linear quadratic problems, Appl. Math. Optim., (2018), 1-29. doi: 10.1007/s00245-018-9513-x.

[15]

J. Wei and T. Wang, Time-consistent mean-variance asset-liability management with random coefficients, Insurance Math. Econom., 77 (2017), 84-96. doi: 10.1016/j.insmatheco.2017.08.011.

[16]

J. Yong, A deterministic linear quadratic time-inconsistent optimal control problem, Math. Control Related Fields, 1 (2011), 83-118. doi: 10.3934/mcrf.2011.1.83.

[17]

J. Yong, Deterministic time-inconsistent optimal control problems--an essentially cooperative approach, Acta Math. Appl. Sin. Engl. Ser., 28 (2012), 1-30. doi: 10.1007/s10255-012-0120-3.

[18]

J. Yong, Time-inconsistent optimal control problem and the equilibrium HJB equation, Math. Control Related Fields, 2 (2012), 271-329. doi: 10.3934/mcrf.2012.2.271.

[19]

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.

[20]

J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations--time-consistent solutions, Trans. Amer. Math. Soc., 369 (2017), 5467-5523. doi: 10.1090/tran/6502.

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