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On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 41.

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This article is concerned with linear quadratic optimal control problems of mean-field stochastic differential equations (MF-SDE) with deterministic coefficients. To treat the time inconsistency of the optimal control problems, linear closed-loop equilibrium strategies are introduced and characterized by variational approach. Our developed methodology drops the delicate convergence procedures in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. When the MF-SDE reduces to SDE, our Riccati system coincides with the analogue in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. However, these two systems are in general different from each other due to the conditional mean-field terms in the MF-SDE. Eventually, the comparisons with pre-committed optimal strategies, open-loop equilibrium strategies are given in details.

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DOI : 10.1051/cocv/2019057
Classification : 93E20, 49N10, 49N70
Keywords: Mean-field linear-quadratic optimal control problems, time inconsistency, closed-loop equilibrium strategies, Riccati system 
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     title = {On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
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Wang, Tianxiao. On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 41. doi : 10.1051/cocv/2019057. http://geodesic.mathdoc.fr/articles/10.1051/cocv/2019057/

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This work is supported in part by NSF of China (Grant 11401404, 11471231, 11231007) and the Fundamental Research Funds for the central Universities (YJ201605).