Filtering of BSDE and FBSDE

  • Guangchen Wang
  • Zhen Wu
  • Jie Xiong
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

In this chapter, we develop some filtering results for the solutions to BSDEs and FBSDEs, which play an important role in studying the optimal control with incomplete information. We first state a theorem on the stochastic filtering of a general stochastic process. The proof of that result can be found in Liptser and Shiyayev [49], so we omit it here. Then, we apply this result to the stochastic filtering for the solutions to BSDEs in Section  3.2 and to those for FBSDEs in Section  3.3.

References

  1. 6.
    Bensoussan, A.: Stochastic Control of Partially Observable Systems. Cambridge University Press, Cambridge (1992)CrossRefGoogle Scholar
  2. 11.
    Bode, H.W., Shannon, C.E.: A simplified derivation of linear least square smoothing and prediction theory. Proc. IRE 38, 417–425 (1950)MathSciNetCrossRefGoogle Scholar
  3. 18.
    Duncan, T.: Doctoral Dissertation. Dept. of Electrical Engineering, Stanford University (1967)Google Scholar
  4. 19.
    El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Finance 7, 1–71 (1997)MathSciNetCrossRefGoogle Scholar
  5. 25.
    Fujisaki, M., Kallianpur, G., Kunita, H.: Stochastic differential equations for the nonlinear filtering problem. Osaka J. Math. 9, 19–40 (1972)MathSciNetMATHGoogle Scholar
  6. 34.
    Kailath, T.: An innovation approach to least-square estimation. Part I: Linear filtering in additive white noise. IEEE Trans. Automat. Control 13, 646–655 (1968)Google Scholar
  7. 35.
    Kailath, T., Frost, P.: An innovation approach to least-square estimation. Part II: Linear smoothing in additive white noise. IEEE Trans. Automat. Control 13, 656–660 (1968)Google Scholar
  8. 40.
    Kushner, H.J.: On the differential equations satisfied by conditional probablitity densities of Markov processes with applications. SIAM J. Control 2, 106–119 (1962)MathSciNetMATHGoogle Scholar
  9. 49.
    Liptser, R.S., Shiryayev, A.N.: Statistics of Random Processes. Springer, New York (1977)CrossRefGoogle Scholar
  10. 56.
    Mortensen, R.E.: Doctoral Dissertation. Dept. of Electrical Engineering. University of California at Berkeley (1966)Google Scholar
  11. 77.
    Stratonovich, R.L.: Conditional Markov processes. Theory Prob. Appl. 5, 156–178 (1960)MathSciNetCrossRefGoogle Scholar
  12. 84.
    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)MathSciNetCrossRefGoogle Scholar
  13. 95.
    Wang, G., Zhang, C., Zhang, W.: Stochastic maximum principle for mean-field type optimal control with partial information. IEEE Trans. Automat. Control 59, 522–528 (2014)MathSciNetCrossRefGoogle Scholar
  14. 104.
    Xiong, J.: An Introduction to Stochastic Filtering Theory. Oxford University Press, London (2008)MATHGoogle Scholar
  15. 109.
    Yong, J., Zhou, X.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999)CrossRefGoogle Scholar
  16. 112.
    Zakai, M.: On the optimal filtering of diffusion processes. Z. Wahrsch. Geb. 11, 230–243 (1969)MathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Guangchen Wang
    • 1
  • Zhen Wu
    • 2
  • Jie Xiong
    • 3
  1. 1.School of Control Science and EngineeringShandong UniversityJinanChina
  2. 2.School of MathematicsShandong UniversityJinanChina
  3. 3.Department of MathematicsSouthern University of Science and TechnologyShenzhenChina

Personalised recommendations