Skip to main content
SpringerLink
Log in

A Maximum Principle for Fully Coupled Forward-Backward Stochastic Control System Driven by Lévy Process with Terminal State Constraints

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

This paper is concerned with a fully coupled forward-backward stochastic optimal control problem where the controlled system is driven by Lévy process, while the forward state is constrained in a convex set at the terminal time. The authors use an equivalent backward formulation to deal with the terminal state constraint, and then obtain a stochastic maximum principle by Ekeland’s variational principle. Finally, the result is applied to the utility optimization problem in a financial market.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Pontryakin L S, Boltyanskti V G, Gamkrelidze R V, et al., The Mathematical Theory of Optimal Control Processes, John Wiley, New York, 1962.

    Google Scholar 

  2. Peng S G, A general maximum principle for optimal control problems, SIAM Journal on Control and Optimization, 1990, 28: 966–979.

    Article  MathSciNet  MATH  Google Scholar 

  3. Peng S G, Backward stochastic differrntial equations and application to optimal control, App. Math. Optim., 1993, 27(4): 125–144.

    Article  Google Scholar 

  4. Wu Z, Maximum principle for optimal control problem of fully coupled forward-backward stochastic control system, Systems Science and Mathematical Sciences, 1998, 11(3): 249–259.

    MathSciNet  MATH  Google Scholar 

  5. Shi J T and Wu Z, The maximum principle for fully coupled forward-backward stochastic control system, Acta Automatica Sinica, 2006, 32(2): 161–169.

    MathSciNet  Google Scholar 

  6. Shi J T and Wu Z, Maximum principle for fully coupled forward-backward stochastic control system with random jumps and applications to finance, Journal of Systems Science & Complexity, 2010, 23(2): 219–231.

    Article  MathSciNet  MATH  Google Scholar 

  7. Meng Q X and Tang M N, Necessary and sufficient conditions for optimal control of stochastic systems associated with Lévy processes, Sci. China, Information Sciences, 2009, 52: 1982–1992.

    Article  MathSciNet  MATH  Google Scholar 

  8. Tang M N and Zhang Q, Optimal variational principle for backward stochastic control systems associated with Lévy processes, Sci. China Mathematics, 2012, 55(4): 745–761.

    Article  MathSciNet  MATH  Google Scholar 

  9. Zhang F, Tang M N, and Meng Q X, Sochastic maximum principle for forward-backward stochastic control systems associated with Lévy processes, Chinese Annals of Mathematics (in China), 2014, 35A(1): 83–100.

    MATH  Google Scholar 

  10. Wang X R and Huang H, Maximum principle for forward-backward stochastic control system driven by Lévy Process, Mathematical Problems in Engineering, 2015, ID 702802, 1–12.

    Google Scholar 

  11. Bielecki T, Jin H, Pliska S, et al., Continuous time mean variance portfolio selection with bankruptcy prohibition, Math. Finance, 2005, 15: 213–244.

    Article  MathSciNet  MATH  Google Scholar 

  12. Ji S L and Zhou X Y, A maximum principle for stochastic optimal control with terminal state constraints, and its appliciations, Communication in Information and Systems, 2003, 6(4): 321–338.

    Article  Google Scholar 

  13. Ji S L and Wei Q M, A maximum principle for fully coupled forward-backward stochastic control systems with terminal state constraints, Journal of Mathematical Analysis and Applications, 2013, 407: 200–210.

    Article  MathSciNet  MATH  Google Scholar 

  14. Nualart D and Schoutens W, Chaotic and predicatable respresentation for Lévy processes, J. Syst. Sci. Comp., 2000, 90: 109–122.

    MATH  Google Scholar 

  15. Nualart D and Schoutens W, BSDE’s and Feynman-Kac formula for Lévy processes with applications in finance, Bernoulli, 2001, 7: 761–776.

    Article  MathSciNet  MATH  Google Scholar 

  16. Protter P, Stochastic Intergration and Differential Equations, Springer-Verlag, Berlin, 1990.

    Book  Google Scholar 

  17. Baghery F, Khelfallah N, Mezerdi B, et al., Fully coupled forward backward stochastic differential equations driven by Lévy processes and application to differential games, De Gruyter, 2014, 22(3): 151–161.

    MATH  Google Scholar 

  18. Mohamed E, Generalized BSDE driven by a Lévy process, Journal of Applied Mathematics and Stochastic Analysis, 2006, ID 85407: 1–25.

    Google Scholar 

  19. Ekeland I, On the variational principle, Journal of Mathematical Analysis & Applications, 1974, 47(2): 324–353.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Financial Engineering, Shandong University of Science and Technology, Qingdao, 266590, China

    Hong Huang, Xiangrong Wang & Meijuan Liu

  2. Institute of Financial Engineering, Shandong Women’s University, Jinan, 250300, China

    Hong Huang

Authors
  1. Hong Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiangrong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Meijuan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiangrong Wang.

Additional information

This research was supported by the National Science Fundation of China under Grant No. 11271007, the National Social Science Fund Project of China under Grant No. 17BGL058, Humanity and Social Science Research Foundation of Ministry of Education of China under Grant No. 15YJA790051.

This paper was recommended for publication by Editor LIU Yungang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, H., Wang, X. & Liu, M. A Maximum Principle for Fully Coupled Forward-Backward Stochastic Control System Driven by Lévy Process with Terminal State Constraints. J Syst Sci Complex 31, 859–874 (2018). https://doi.org/10.1007/s11424-017-6209-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-017-6209-2

Keywords

Search

Navigation