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An Indefinite Stochastic Linear Quadratic Optimal Control Problem with Delay and Related Forward–Backward Stochastic Differential Equations

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Abstract

In this paper, we will study an indefinite stochastic linear quadratic optimal control problem, where the controlled system is described by a stochastic differential equation with delay. By introducing the relaxed compensator as a novel method, we obtain the well-posedness of this linear quadratic problem for indefinite case. And then, we discuss the uniqueness and existence of the solutions for a kind of anticipated forward–backward stochastic differential delayed equations. Based on this, we derive the solvability of the corresponding stochastic Hamiltonian systems, and give the explicit representation of the optimal control for the linear quadratic problem with delay in an open-loop form. The theoretical results are validated as well on the control problems of engineering and economics under indefinite condition.

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Acknowledgements

Na Li is supported by the National Natural Science Foundation of China (Grant No. 11626142), the Natural Science Foundation of Shandong Province (Grant No. ZR2016AB08), the Colleges and Universities Science and Technology Plan Project of Shandong Province (Grant No. J16LI55), Key Laboratory of Oceanographic Big Data Mining and Application of Zhejiang Province (OBDMA201604), and the Fostering Project of Dominant Discipline and Talent Team of Shandong University of Finance and Economics. Zhen Wu is supported by the National Natural Science Foundation of China (Grant No. 61573217), the National High-level personnel of special support program and the Chang Jiang Scholar Program of Chinese Education Ministry. The authors would like to thank the associated editor and the anonymous referees for their constructive and insightful comments to improve the quality of this work. And we would like to thank Prof. Zhiyong Yu for many helpful discussions and suggestions.

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Authors and Affiliations

  1. School of Statistics, Shandong University of Finance and Economics, Jinan, Shandong, China

    Na Li

  2. School of Mathematics, Shandong University, Jinan, Shandong, China

    Yuan Wang & Zhen Wu

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  1. Na Li
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  2. Yuan Wang
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  3. Zhen Wu
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Correspondence to Zhen Wu.

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Li, N., Wang, Y. & Wu, Z. An Indefinite Stochastic Linear Quadratic Optimal Control Problem with Delay and Related Forward–Backward Stochastic Differential Equations. J Optim Theory Appl 179, 722–744 (2018). https://doi.org/10.1007/s10957-018-1237-1

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  • DOI: https://doi.org/10.1007/s10957-018-1237-1

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