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



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.


Stochastic linear quadratic problem Stochastic differential delayed equations Forward–backward stochastic differential equations Hamiltonian system 

Mathematics Subject Classification

93E20 60H10 49N10 



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.


  1. 1.
    Mohammed, S.E.A.: Stochastic differential equations with memory: theory, examples and applications. Ustunel Prog. Prob. Birkhauser 42, 1–77 (1998)MATHGoogle Scholar
  2. 2.
    Arriojas, M., Hu, Y., Mohammed, S.E.A., Pap, G.: A delayed black and scholes formula. Stoch. Anal. Appl. 25(2), 471–492 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chen, L., Wu, Z.: Maximum principle for the stochastic optimal control problem with delay and application. Automatica 46(6), 1074–1080 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Peng, S., Yang, Z.: Anticipated backward stochastic differential equations. Ann. Prob. 37(3), 877–902 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Mohammed, S.E.A.: Stochastic functional differential equations. Pitman Adv. Pub. 18(2), 63–64 (1984)Google Scholar
  6. 6.
    Richard, J.P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39(10), 1667–1694 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Swishchuk, A.: Modeling and pricing of variance swaps for stochastic volatilities with delay. Wilmott Mag. 19(9), 63–73 (2005)Google Scholar
  8. 8.
    Øksendal, B., Sulem, A., Zhang, T.: Optimal control of stochastic delayed equations and time-advanced backward stochastic differential equations. Adv. Appl. Prob. 43(2), 572–596 (2011)CrossRefMATHGoogle Scholar
  9. 9.
    Lv, S., Tao, R., Wu, Z.: Maximum principle for optimal control of anticipated forward–backward stochastic differential delayed systems with regime switching. Optimal Control Appl. Methods 37(1), 154–175 (2016)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Yu, Z.: The stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls. Automatica 48(10), 2420–2432 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Peng, S.: New development in stochastic maximum principle and related backward stochastic differential equations. In: Proceedings of 31st CDC Conference, Tucson (1992)Google Scholar
  12. 12.
    Chen, S., Li, X., Zhou, X.Y.: Stochastic linear quadratic regulators with indefinite control weight costs. SIAM J. Control Optim. 36(5), 1685–1702 (1998)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Rami, M.A., Zhou, X.Y., Moore, J.B.: Well-posedness and attainability of indefinite stochastic linear quadratic control in infinite time horizon. Syst. Control Lett. 41(2), 123–133 (2000)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Rami, M., Chen, X., Moore, J., Zhou, X.: Solvability and asymptotic behavior of generalized Riccati equations arising in indefinite stochastic LQ controls. IEEE Trans. Autom. Control 46(3), 428–440 (2001)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Qian, Z., Zhou, X.Y.: Existence of solutions to a class of indefinite stochastic Riccati equations. SIAM J. Control Optim. 51(1), 221–229 (2013)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ivanov, I.: The LMI approach for stabilizing of linear stochastic systems. Int. J. Stoch. Anal. 2013, 1–5 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Yu, Z.: Equivalent cost functionals and stochastic linear quadratic optimal control problems. ESAIM Control Optim. Calc. Var. 19(1), 78–90 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Huang, J., Yu, Z.: Solvability of indefinite stochastic Riccati equations and linear quadratic optimal control problems. Syst. Control Lett. 68(1), 68–75 (2014)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Li, N., Wu, Z., Yu, Z.: An indefinite stochastic linear quadratic optimal control problem for the FBSDE system with jumps. In: The 34th Chinese Control Conference, pp. 1682–1686 (2015)Google Scholar
  20. 20.
    Li, N., Wu, Z., Yu, Z.: Indefinite stochastic linear-quadratic optimal control problems with random jumps and related stochastic Riccati equations. Sci. China Math. (2017). Google Scholar
  21. 21.
    Hu, Y., Peng, S.: Solution of forward–backward stochastic differential equations. Prob. Theory Relat. Fields 103(2), 273–283 (1995)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Peng, S., Wu, Z.: Fully coupled forward–backward stochastic differential equations and applications to optimal control. SIAM J. Control Optim. 37(3), 825–843 (1999)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Zhang, H., Li, L., Xu, J., Fu, M.: Linear quadratic regulation and stabilization of discrete-time systems with delay and multiplicative noise. IEEE Trans. Autom. Control 60(10), 2599–2613 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Zhang, H., Xu, J.: Control for Itô stochastic systems with input delay. IEEE Trans. Autom. Control 62(1), 350–365 (2017)CrossRefMATHGoogle Scholar
  25. 25.
    Ramsey, F.P.: A mathematical theory of saving. Econ. J. 38(152), 543–559 (1928)CrossRefGoogle Scholar
  26. 26.
    Huang, J., Wang, G., Wu, Z.: Optimal premium policy of an insurance firm: full and partial information. Insur. Math. Econ. 47, 208–215 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of StatisticsShandong University of Finance and EconomicsJinanChina
  2. 2.School of MathematicsShandong UniversityJinanChina

Personalised recommendations