Skip to main content
SpringerLink
Log in

LQ Optimal Control Models with Incomplete Information

  • Chapter
  • First Online:
An Introduction to Optimal Control of FBSDE with Incomplete Information

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

  • 972 Accesses

Abstract

In this chapter, we consider the so-called LQ problem with incomplete information aiming at obtaining more explicit results comparing with those of the previous chapters. We first consider this problem when the state is given by a linear FBSDE. After that we will specialize our results to the case when the state is governed by a BSDE only. In this case, explicit solution will be presented. Finally, we will apply our results to an optimal premium problem.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 19.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 29.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bellman, R., Glicksberg, I., Gross, O.: Some Aspects of the Mathematical Theory of Control Processes. Rand Corporation, Santa Monica (1958)

    MATH  Google Scholar 

  2. Bensoussan, A.: Stochastic Control by Functional Analysis Methods. North-Holland Publishing Company, New York (1982)

    MATH  Google Scholar 

  3. Bensoussan, A., Viot, M.: Optimal control of stochastic linear distributed parameter systems. SIAM J. Control 13, 904–926 (1975)

    Article  MathSciNet  Google Scholar 

  4. Dokuchaev, N.G., Zhou, X.: Stochastic control problems with terminal contingent conditions. J. Math. Anal. Appl. 238, 143–165 (1999)

    Article  MathSciNet  Google Scholar 

  5. El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Finance 7, 1–71 (1997)

    Article  MathSciNet  Google Scholar 

  6. Huang, J., Wang, G., Xiong, J.: A maximum principle for partial information backward stochastic control problems with applications. SIAM J. Control Optim. 48, 2106–2117 (2009)

    Article  MathSciNet  Google Scholar 

  7. Kalman, R.E.: Contributions to the theory of optimal control. Bol. Soc. Mat. Mexicana 5, 102–119 (1960)

    MathSciNet  MATH  Google Scholar 

  8. Kushner, H.J.: Optimal stochastic control. IRE Trans. Automat. Control 7, 120–122 (1962)

    Article  Google Scholar 

  9. Letov, A.M.: Analytic design of regulators. Avtomat. Telemekh. 436–446, 561–571, 661–669 (1960, in Russian); English Transl.: Automat. Remote Control 21 (1960)

    Google Scholar 

  10. Li, N., Yu, Z.: Recursive stochastic linear-quadratic optimal control and nonzero-sum differential game problems with random jumps. Adv. Differ. Equ. 144 (2015). https://doi.org/10.1186/s13662-015-0439-1

  11. Lim, A.E.B., Zhou, X.: Linear-quadratic control of backward stochastic differential equations. SIAM J. Control Optim. 40, 450–474 (2001)

    Article  MathSciNet  Google Scholar 

  12. Meng, Q.: Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems with partial information. Sci. China Ser. A Math. 52, 1579–1588 (2009)

    Article  MathSciNet  Google Scholar 

  13. Økesandal, B., Sulem, A.: Maximum principle for optimal control of stochastic differential equations with applications. SIAM J. Control Optim. 48, 2945–2976 (2010)

    Article  Google Scholar 

  14. Shi, J., Wang, G., Xiong, J.: Leader-follower stochastic differential game with asymmetric information and applications. Automat. J. IFAC 63, 60–73 (2016)

    Article  MathSciNet  Google Scholar 

  15. Shi, J., Wang, G., Xiong, J.: Linear-quadratic stochastic Stackelberg differential game with asymmetric information. Sci. China Inform. Sci. 60, 092202:1–092202:15 (2017)

    Google Scholar 

  16. Shi, J., Wu, Z.: The maximum principle for fully coupled forward-backward stochastic control system. Acta Automat. Sinica 32, 161–169 (2006)

    MathSciNet  Google Scholar 

  17. Tang, S.: General linear quadratic optimal stochastic control problems with random coefficients: linear stochastic Hamilton systems and backward stochastic Riccati equations. SIAM J. Control Optim. 42, 53–75 (2003)

    Article  MathSciNet  Google Scholar 

  18. Wang, X., Gao, Z., Wu, Z.: Forward-backward stochastic differential equation and the linear quadratic stochastic optimal control. Acta Automat. Sinica 29, 32–37 (2003)

    MathSciNet  Google Scholar 

  19. Wang, G., Wu, Z.: Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems. J. Math. Anal. Appl. 342, 1280–1296 (2008)

    Article  MathSciNet  Google Scholar 

  20. Wang, G., Wu, Z., Xiong, J.: Maximum principles for forward-backward stochastic control systems with correlated state and observation noises. SIAM J. Control Optim. 51, 491–524 (2013)

    Article  MathSciNet  Google Scholar 

  21. Wang, G., Wu, Z., Xiong, J.: A linear-quadratic optimal control problem of forward-backward stochastic differential equations with partial information. IEEE Trans. Automat. Control 60, 2904–2916 (2015)

    Article  MathSciNet  Google Scholar 

  22. Wang, G., Xiao, H., Xing, G.: An optimal control problem for mean-field forward-backward stochastic differential equation with noisy. Automatica J. IFAC 86, 104–109 (2017)

    Article  MathSciNet  Google Scholar 

  23. Wonham, W.M.: On the separation theorem of stochastic control. SIAM J. Control 6, 312–326 (1968)

    Article  MathSciNet  Google Scholar 

  24. Wonham, W.M.: On a matrix Riccati equation of stochastic control. SIAM J. Control 6, 681–697 (1968)

    Article  MathSciNet  Google Scholar 

  25. Xiao, H.: The maximum principle for partially observed optimal control of forward-backward stochastic systems with random jumps. J. Syst. Sci. Complex 24, 1083–1099 (2011)

    Article  MathSciNet  Google Scholar 

  26. Yong, J.: Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions. SIAM J. Control Optim. 48, 4119–4156 (2010)

    Article  MathSciNet  Google Scholar 

  27. Yong, J., Zhou, X.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999)

    Book  Google Scholar 

  28. Yu, Z.: Linear-quadratic optimal control and nonzero-sum differential game of forward-backward stochastic system. Asian J. Control 14, 173–185 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Control Science and Engineering, Shandong University, Jinan, China

    Guangchen Wang

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

    Zhen Wu

  3. Department of Mathematics, Southern University of Science and Technology, Shenzhen, China

    Jie Xiong

Authors
  1. Guangchen Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhen Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jie Xiong
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2018 The Author(s), under exclusive licence to Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Wang, G., Wu, Z., Xiong, J. (2018). LQ Optimal Control Models with Incomplete Information. In: An Introduction to Optimal Control of FBSDE with Incomplete Information. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-79039-8_5

Download citation

Publish with us

Policies and ethics

Search

Navigation