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Linear Quadratic Leader-Follower Stochastic Differential Games: Closed-Loop Solvability

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Abstract

In this paper, a leader-follower stochastic differential game is studied for a linear stochastic differential equation with quadratic cost functionals. The coefficients in the state equation and the weighting matrices in the cost functionals are all deterministic. Closed-loop strategies are introduced, which require to be independent of initial states; and such a nature makes it very useful and convenient in applications. The follower first solves a stochastic linear quadratic optimal control problem, and his optimal closed-loop strategy is characterized by a Riccati equation, together with an adapted solution to a linear backward stochastic differential equation. Then the leader turns to solve a stochastic linear quadratic optimal control problem of a forward-backward stochastic differential equation, necessary conditions for the existence of the optimal closed-loop strategy for the leader is given by a Riccati equation. Some examples are also given.

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Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    Zixuan Li & Jingtao Shi

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  1. Zixuan Li
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  2. Jingtao Shi
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Correspondence to Jingtao Shi.

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The authors declare no conflict of interest.

Additional information

This work was supported by National Key Research & Development Program of China under Grant No. 2022YFA1006104, National Natural Science Foundations of China under Grant Nos. 11971266, 11831010, and Shandong Provincial Natural Science Foundations under Grant Nos. ZR2022JQ01, ZR2020ZD24, ZR2019ZD42.

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Li, Z., Shi, J. Linear Quadratic Leader-Follower Stochastic Differential Games: Closed-Loop Solvability. J Syst Sci Complex 36, 1373–1406 (2023). https://doi.org/10.1007/s11424-023-1261-6

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  • DOI: https://doi.org/10.1007/s11424-023-1261-6

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