Skip to main content
SpringerLink
Log in

Near-optimality conditions in stochastic control of linear fully coupled FBSDEs

  • Published:
Afrika Matematika Aims and scope Submit manuscript

Abstract

In this paper, we deal with near optimality for stochastic control problems where the controlled system is described by a linear fully coupled forward–backward stochastic differential equation. We assume that the forward diffusion coefficient depends explicitly on the control variable and the control domain is not necessarily convex. We establish necessary as well as sufficient conditions for near optimality, satisfied by all near optimal controls. The proof of the main result is based on Ekeland’s variational principle and some estimates on the state and the adjoint processes with respect to the control variable. Finally an example which illustrate our results is presented.

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. Antonelli, F.: Backward–forward stochastic differential equation. Ann. Appl. Prob. 3, 777–793 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bahlali, K., Gherbal, B., Mezerdi, B.: Existence of optimal solutions in the control of FBSDEs. Syst. Control Lett. 60(5), 344–349 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bahlali, K., Gherbal, B., Mezerdi, B.: Existence and optimality conditions in the control of linear FBSDEs. Rand. Oper. Stoch. Equ. 18, 185–197 (2010)

    MathSciNet  MATH  Google Scholar 

  4. Bahlali, K., Khelfallah, N., Mezerdi, B.: Necessary and sufficient conditions for near-optimality in stochastic control of FBSDEs. Syst. Control Lett. 58, 857–864 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bahlali, K., Mezerdi, B., Ouknine, Y.: The maximum principle for optimal control of diffusions with non-smooth coefficients. Stochast. Stochast. Rep. 57, 303–316 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chighoub, F., Mezerdi, B.: Near optimality conditions in stochastic control of jump diffusion processes. Syst. Control Lett. 60, 907–916 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Clarke, F.H., Stern, R.J. (eds.): Nonlinear Analysis, Differential Equations and Control, pp. 503–549. Kluwer Academic Publishers, Dordrecht (1999)

    Google Scholar 

  8. Clarke, F.H., Stern, R.J. (eds.): Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Ekeland, I.: Non convex minimisation problems. Bull. Am. Math. Soc. NS 1, 324–353 (1979)

    MathSciNet  Google Scholar 

  11. Elliott, R.J., Kohlmann, M.: The variational principle and stochastic optimal control. Stochastics 3, 229–241 (1980)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Hamadène, S.: Backward–forward SDE’s and stochastic differential game. Stoch. Proc. Appl. 77, 1–15 (1998)

    Article  MATH  Google Scholar 

  14. Huang, J., Li, X., Wang, G.: Near optimal control problems for linear forward backward stochastic systems. Automatica 46, 397–404 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hui, E., Huang, J., Li, X., Wang, G.: Near optimal control for stochastic recursive problems. Syst. Control Lett. 60, 161–168 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hu, Y., Peng, S.: Solution of forward–backward stochastic differential equations. Prob. Theory Relat. Fields 103, 273–283 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ma, J., Protter, P., Yong, J.: Solving forward–backward stochastic differential equations explicitly: a four step scheme. Prob. Theory Relat. Fields 98, 339–359 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mezerdi, B.: Necessary conditions for optimality for a diffusion with non smooth drift. Stochastics 24, 305–326 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  19. Øksendal, B., Sulem, A.: Maximum principles for optimal control of forward-backward stochastic differential equations with jumps. SIAM J. Control Optim. 48(5), 2945–2976 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55–61 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Peng, S.: Backward stochastic differential equation and application to optimal control. Appl. Math. Optim. 27, 125–144 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  22. Peng, S., Wu, Z.: Fully coupled forward backward stochastic differential equations and application to optimal control. SIAM J. Control Optim. 37(3), 825–843 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tang, M.: Stochastic maximum principle of near-optimal control of fully coupled forward-backward stochastic differential equation. Abstr. Appl. Anal. 2014, 361259 (2014). doi:10.1155/2014/361259

  24. Wu, Z.: Maximum principle for optimal control problem of fully coupled forward–backward stochastic systems. Syst. Sci. Math. Sci 11(3), 249–259 (1998)

    MathSciNet  MATH  Google Scholar 

  25. Xu, W.: Stochastic maximum principale for optimal control problem of forward and backward. J. Aust. Math. Soc. Ser. B 37, 172–1785 (1995)

    Article  MATH  Google Scholar 

  26. Yong, J.: Optimality variational principle for controlled forward–backward stochastic differential equations with mixed initial-terminal conditions. SIAM J. Cont. Optim. 48(6), 4119–4156 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yong, J., Zhou, X.Y.: Stochastic controls, Hamiltonian Systems and HJB Equations. Springer Verlag, Berlin (1999)

    MATH  Google Scholar 

  28. Zhou, X.Y.: Stochastic near-optimal controls: necessary and sufficient conditions for near-optimality. SIAM.J. Control Optim 36(3), 929–947 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory of Applied Mathematics, University Med Khider, PO. Box 145, 07000, Biskra, Algeria

    Nabil Khelfallah & Brahim Mezerdi

Authors
  1. Nabil Khelfallah
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Brahim Mezerdi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Brahim Mezerdi.

Additional information

Partially supported by French Algerian Program, PHC Tassili 13 MDU 887.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khelfallah, N., Mezerdi, B. Near-optimality conditions in stochastic control of linear fully coupled FBSDEs. Afr. Mat. 27, 327–343 (2016). https://doi.org/10.1007/s13370-015-0346-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13370-015-0346-3

Keywords

Search

Navigation