Skip to main content
SpringerLink
Log in

Stochastic Maximum Principle for Controlled Backward Delayed System via Advanced Stochastic Differential Equation

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The main contributions of this paper are three old. First, our primary concern is to investigate a class of stochastic recursive delayed control problems that naturally arise with strong backgrounds but have not been well studied yet. For illustration, some concrete examples are provided here. Second, it is interesting that a new class of time-advanced stochastic differential equations (ASDEs) is introduced as the adjoint process via duality relation. To our knowledge, such equations have never been discussed in literature, although they have considerable research values. An existence and uniqueness result for ASDEs is presented. Third, to illustrate our theoretical results, some dynamic optimization problems are discussed based on our stochastic maximum principles. It is interesting that the optimal controls are derived explicitly by solving the associated time-advanced ordinary differential equation (AODE), the counterpart of the ASDE in its deterministic setup.

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. Mohammed, S.E.A.: Stochastic Functional Differential Equations. Pitman Advanced Publishing Program, Boston (1984)

    MATH  Google Scholar 

  2. Mohammed, S.E.A.: Stochastic Differential Equations with Memory: Theory, Examples and Applications. Stochastic Analysis and Related Topics 6, Progress in Probability. Birkhäuser, Basel (1996)

    Google Scholar 

  3. Peng, S., Yang, Z.: Anticipated backward stochastic differential equation. Ann. Probab. 37, 877–902 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Delong, Ł., Imkeller, P.: Backward stochastic differential equations with time delayed generators-results and counterexamples. Ann. Appl. Probab. 20, 1512–1536 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Delong, Ł.: BSDEs with time-delayed generators of a moving average type with applications to non-monotone preferences. Stoch. Models 28, 281–315 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Delong, Ł.: Applications of time-delayed backward stochastic differential equations to pricing, hedging and portfolio management. Working paper (2012)

  7. Øksendal, B., Sulem, A.: A maximum principle for optimal control of stochastic systems with delay, with applications to finance. In: Menaldi, J.L., Rofman, E., Sulem, A. (eds.) Optimal Control and Partial Differential Equations, pp. 64–79. IOS Press, Amsterdam (2001)

    Google Scholar 

  8. Øksendal, B., Sulem, A., Zhang, T.: Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations. Adv. Appl. Probab. 43, 572–596 (2011)

    Article  Google Scholar 

  9. Antonellia, F., Baruccib, E., Mancinoc, M.E.: Asset pricing with a forward-backward stochastic differential utility. Econ. Lett. 72, 151–157 (2001)

    Article  Google Scholar 

  10. Federico, S.: A stochastic control problem with delay arising in a pension fund model. Finance Stoch. 15, 421–459 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen, L., Wu, Z.: Maximum principle for the stochastic optimal control problem with delay and application. Automatica 46, 1074–1080 (2010)

    Article  MATH  Google Scholar 

  12. Larssen, B.: Dynamic programming in stochastic control of systems with delay. Stoch. Stoch. Rep. 74, 651–673 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kohlmann, M., Zhou, X.Y.: Relationship between backward stochastic differential equations and stochastic controls: a linear-quadratic approach. SIAM J. Control Optim. 38, 1392–1407 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ma, J., Yong, J.: Forward–Backward Stochastic Differential Equations and Their Applications. Lecture Notes in Math., vol. 1702. Springer, Berlin (1999)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. El Karoui, N., Peng, S., Quenez, M.C.: A dynamic maximum principle for the optimization of recursive utilities under constrains. Ann. Appl. Probab. 11, 664–693 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Augustynowicz, A., Leszczynski, H., Walter, W.: On some nonlinear ordinary differential equations with advanced arguments. Nonlinear Anal. 53, 495–505 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Cooke, K.L., Wiener, J.: An equation alternately to retarded and advanced type. Proc. Am. Math. Soc. 99, 726–732 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hall, A.J., Wake, G.C., Gandar, P.W.: Steady size distributions for cells in one dimensional plant issues. J. Math. Biol. 30, 101–123 (1991)

    Article  MATH  Google Scholar 

  21. Kato, T., McLeod, J.B.: The functional-differential equation. Bull. Am. Math. Soc. 77, 891–937 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  22. Papavassilopoulos, G.P., Olsder, G.J.: On a linear differential equation of the advanced type. J. Math. Anal. Appl. 103, 74–82 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  23. Yoneda, T.: On the functional-differential equation of advanced type. J. Math. Anal. Appl. 332, 487–496 (2006)

    Article  MathSciNet  Google Scholar 

  24. Cadenillas, A., Karatzas, I.: The stochastic maximum principle for linear convex optimal control with random coefficients. SIAM J. Control Optim. 33, 590–624 (1995)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author acknowledges the support from the Fundamental Research Funds for the Central Universities (2010QS05), P.R. China. The first author also thanks Department of Applied Mathematics, The Hong Kong Polytechnic University for their hospitality during her visit to Hong Kong. The second author acknowledges the support of RGC Earmarked grant 500909 and research fund of Hong Kong Polytechnic University (A-PL14).

Author information

Authors and Affiliations

  1. Department of Mathematics, China University of Mining and Technology, Beijing, 100083, China

    Li Chen

  2. Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, Hong Kong

    Jianhui Huang

Authors
  1. Li Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianhui Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianhui Huang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, L., Huang, J. Stochastic Maximum Principle for Controlled Backward Delayed System via Advanced Stochastic Differential Equation. J Optim Theory Appl 167, 1112–1135 (2015). https://doi.org/10.1007/s10957-013-0386-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-013-0386-5

Keywords

Search

Navigation