Skip to main content
SpringerLink
Log in

The maximum principle for partially observed optimal control of forward-backward stochastic systems with random jumps

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

Abstract

This paper studies the problem of partially observed optimal control for forward-backward stochastic systems which are driven both by Brownian motions and an independent Poisson random measure. Combining forward-backward stochastic differential equation theory with certain classical convex variational techniques, the necessary maximum principle is proved for the partially observed optimal control, where the control domain is a nonempty convex set. Under certain convexity assumptions, the author also gives the sufficient conditions of an optimal control for the aforementioned optimal optimal problem. To illustrate the theoretical result, the author also works out an example of partial information linear-quadratic optimal control, and finds an explicit expression of the corresponding optimal control by applying the necessary and sufficient maximum principle.

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. N. Williams, On dynamic principal-agent problems in continuous time, working paper, 2009.

  2. R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC, 2004.

  3. A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, 1992.

  4. S. Liptser and N. Shiryaev, Statistics of Random Processes, I, II, Springer-verlag, 1977.

  5. B. Øksendal and A. Sulem, Maximum principles for optimal control of forward-backward stochastic differential equations with jumps, SIAM Journal on Control and Optimization, 2010, 48(5): 2945–2976.

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. S. Tang, The maximum principle for partially observed optimal control of stochastic differential equations, SIAM Journal on Control and Optimization, 1998, 36: 1596–1617.

    Article  MATH  MathSciNet  Google Scholar 

  8. G. Wang and Z. Wu, Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems, Journal of Mathematical Analysis and Applications, 2008, 342: 1280–1296.

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Wang and Z. Wu, The maximum principles for stochastic recursive optimal control problems under partial information, IEEE Transactions on Automatic control, 2009, 54(6): 1230–1242.

    Article  Google Scholar 

  10. Z. Wu, A maximum principle for partially observed optimal control of forward-backward stochastic systems, Science in China, Series F, 2010, 53(11): 2205–2214.

    Article  MATH  Google Scholar 

  11. G. Barles, R. Buckdahn, and E. Pardoux, Backward stochastic differential equations and integralpartial differential equations, Stochastics, 1997, 60: 7–83.

    MathSciNet  Google Scholar 

  12. S. Tang and X. Li, Necessary contions for optimal control of stochastic systems with random jumps, SIAM Journal on Control and Optimization, 1994, 32(5): 1447–1475.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. Huang, X. Li, and G. Wang, Maximum principles for a class of partial information risk-sensitive optimal controls, IEEE Transaction on Automatic Control, 2010, 55(6): 1438–1443.

    Article  MathSciNet  Google Scholar 

  14. Z. Wu, Forward-backward stochastic differential equations, linear quadratic stochastic optimal control and nonzero sum differential games, Journal of Systems Science & Complexity, 2005, 18(2): 179–192.

    MATH  Google Scholar 

  15. N. El-Karoui, S. Peng, and M. Quenze, Backward stochastic differential equations in finance, Mathematical Finance, 1997, 7(1): 1–71.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University at Weihai, Weihai, 264209, China

    Hua Xiao

  2. School of Mathematics, Shandong University, Jinan, 250100, China

    Hua Xiao

Authors
  1. Hua Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hua Xiao.

Additional information

This research is supported by the National Nature Science Foundation of China under Grant Nos 11001156, 11071144, the Nature Science Foundation of Shandong Province (ZR2009AQ017), and Independent Innovation Foundation of Shandong University (IIFSDU), China.

This paper was recommended for publication by Editor Jifeng ZHANG.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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). https://doi.org/10.1007/s11424-011-9311-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-011-9311-x

Key words

Search

Navigation