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Linear quadratic control of backward stochastic differential equation with partial information

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  • Wang, Guangchen
  • Wang, Wencan
  • Yan, Zhiguo

Abstract

In this paper, we study an optimal control problem of linear backward stochastic differential equation (BSDE) with quadratic cost functional under partial information. This problem is solved completely and explicitly by using a stochastic maximum principle and a decoupling technique. In terms of the maximum principle, a stochastic Hamiltonian system, which is a forward-backward stochastic differential equation (FBSDE) with filtering, is obtained. By decoupling the stochastic Hamiltonian system, three Riccati equations, a BSDE with filtering, and a stochastic differential equation (SDE) with filtering are derived. We then get a feedback representation of optimal control. An explicit formula for the corresponding optimal cost is also established. As illustrative examples, we consider two special scalar-valued control problems and give some numerical simulations.

Suggested Citation

  • Wang, Guangchen & Wang, Wencan & Yan, Zhiguo, 2021. "Linear quadratic control of backward stochastic differential equation with partial information," Applied Mathematics and Computation, Elsevier, vol. 403(C).
    • Handle: RePEc:eee:apmaco:v:403:y:2021:i:c:s009630032100254x
    DOI: 10.1016/j.amc.2021.126164
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    References listed on IDEAS

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    1. J. T. Shi & Z. Wu, 2010. "Maximum Principle for Partially-Observed Optimal Control of Fully-Coupled Forward-Backward Stochastic Systems," Journal of Optimization Theory and Applications, Springer, vol. 145(3), pages 543-578, June.
    2. Wu, Zhen & Zhuang, Yi, 2018. "Linear-quadratic partially observed forward–backward stochastic differential games and its application in finance," Applied Mathematics and Computation, Elsevier, vol. 321(C), pages 577-592.
    3. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    4. Xiong, Jie, 2008. "An Introduction to Stochastic Filtering Theory," OUP Catalogue, Oxford University Press, number 9780199219704.
    5. Kai Du & Zhen Wu, 2019. "Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential System and Application," Mathematical Problems in Engineering, Hindawi, vol. 2019, pages 1-17, February.
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    Cited by:

    1. Fikriye Yılmaz & Hacer Öz Bakan & Gerhard-Wilhelm Weber, 2024. "Weak Second-Order Conditions of Runge–Kutta Method for Stochastic Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 497-517, July.
    2. Huang, Pengyan & Wang, Guangchen & Wang, Wencan & Wang, Yu, 2023. "A linear-quadratic mean-field game of backward stochastic differential equation with partial information and common noise," Applied Mathematics and Computation, Elsevier, vol. 446(C).
    3. Wang, Yu & Yan, Zhiguo, 2023. "Pareto-based Stackelberg differential game for stochastic systems with multi-followers," Applied Mathematics and Computation, Elsevier, vol. 436(C).

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