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

Author

Listed:
  • Na Li

    (Shandong University of Finance and Economics)

  • Yuan Wang

    (Shandong University)

  • Zhen Wu

    (Shandong University)

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.

Suggested Citation

  • Na Li & Yuan Wang & Zhen Wu, 2018. "An Indefinite Stochastic Linear Quadratic Optimal Control Problem with Delay and Related Forward–Backward Stochastic Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 722-744, November.
    • Handle: RePEc:spr:joptap:v:179:y:2018:i:2:d:10.1007_s10957-018-1237-1
    DOI: 10.1007/s10957-018-1237-1
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    References listed on IDEAS

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    1. Huang, Jianhui & Wang, Guangchen & Wu, Zhen, 2010. "Optimal premium policy of an insurance firm: Full and partial information," Insurance: Mathematics and Economics, Elsevier, vol. 47(2), pages 208-215, October.
    2. Ivan Ivanov, 2013. "The LMI Approach for Stabilizing of Linear Stochastic Systems," International Journal of Stochastic Analysis, Hindawi, vol. 2013, pages 1-5, August.
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    Cited by:

    1. Peng Wang, 2023. "Risk-Sensitive Maximum Principle for Controlled System with Delay," Mathematics, MDPI, vol. 11(4), pages 1-12, February.
    2. Dela Vega, Engel John C. & Elliott, Robert J., 2022. "Backward stochastic differential equations with regime-switching and sublinear expectations," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 278-298.
    3. Li Chen & Peipei Zhou & Hua Xiao, 2023. "Backward Stackelberg Games with Delay and Related Forward–Backward Stochastic Differential Equations," Mathematics, MDPI, vol. 11(13), pages 1-18, June.

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