Journal of Optimization Theory and Applications

, Volume 178, Issue 2, pp 363–382 | Cite as

Near-Optimal Control of Stochastic Recursive Systems Via Viscosity Solution

  • Liangquan ZhangEmail author
  • Qing Zhou


In this paper, we study the near-optimal control for systems governed by forward–backward stochastic differential equations via dynamic programming principle. Since the nonsmoothness is inherent in this field, the viscosity solution approach is employed to investigate the relationships among the value function, the adjoint equations along near-optimal trajectories. Unlike the classical case, the definition of viscosity solution contains a perturbation factor, through which the illusory differentiability conditions on the value function are dispensed properly. Moreover, we establish new relationships between variational equations and adjoint equations. As an application, a kind of stochastic recursive near-optimal control problem is given to illustrate our theoretical results.


Dynamic programming principle Forward–backward stochastic differential equations Near-optimal control Super-/subdifferentials 

Mathematics Subject Classification

93E20 49L20 



The authors wish to thank the editor and the referees for their valuable comments and constructive suggestions which improved the presentation of this manuscript. We also thank Dr. J. Yang for her careful reading and suggestions. L. Zhang acknowledges the financial support partly by the National Nature Science Foundation of China (Nos. 11701040, 11471051 and 11371362) and Innovation Foundation of BUPT for Youth (No. 500417024). Q. Zhou acknowledges the financial support partly by the National Nature Science Foundation of China (Nos. 11471051 and 11371362).


  1. 1.
    Bismut, J.M.: Théorie Probabiliste du Contrôle des Diffusions. Memoirs of the American Mathematical Society, 176, Providence, Rhode Island (1973)Google Scholar
  2. 2.
    Pardoux, E., Peng, S.G.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 55–61 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Duffie, D., Epstein, L.G.: Stochastic differential utility. Econometrica 60(2), 353–394 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    El Karoui, N., Peng, S.G., Quenez, M.C.: A dynamic maximum principle for the optimization of recursive utilities under constraints. Ann. Appl. Probab. 11(3), 664–693 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Peng, S.G., Wu, Z.: Fully coupled forward–backward stochastic differential equations and applications to optimal control. SIAM J. Control Optim. 37(3), 825–843 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Pardoux, E., Tang, S.J.: Forward–backward stochastic differential equations and quasilinear parabolic PDEs. Probab. Theory Relat. Fields 114(2), 123–150 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ma, J., Yong, J.M.: Forward–Backward Stochastic Differential Equations and their Applications. Springer, New York (1999)zbMATHGoogle Scholar
  8. 8.
    Zhou, X.Y., Sethi, S.: A sufficient condition for near optimal stochastic controls and its applications to manufacturing systems. Appl. Math. Optim. 29, 67–92 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chighoub, F., Mezerdi, B.: Near optimality conditions in stochastic control of jump diffusion processes. Syst. Control Lett. 60, 907–916 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lin, X.Y., Zhang, W.H.: A maximum principle for optimal control of discrete-time stochastic systems with multiplicative noise. IEEE Trans. Autom. Control 60(4), 1121–1126 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Meng, Q.X., Shen, Y.: A revisit to stochastic near-optimal controls: the critical case. Syst. Control Lett. 82, 79–85 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liu, Y., Yin, G., Zhou, X.Y.: Near-optimal controls of random-switching LQ problems with indefinite control weight costs. Automatica 41, 1063–1070 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Zhou, X.Y.: Deterministic near-optimal controls, part I: necessary and sufficient conditions for near-optimality. J. Optim. Theory Appl. 85, 473–488 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Zhou, X.Y.: Deterministic near-optimal controls, part II: dynamic programming and viscosity solution approach. Math. Oper. Res. 21, 655–674 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Zhou, X.Y.: Stochastic near-optimal controls: necessary and sufficient conditions for near-optimality. SIAM J. Control Optim. 39, 929–947 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zhang, W.H., Lin, X.Y., Chen, B.: LaSalle-type theorem and its applications to infinite horizon optimal control of discrete-time nonlinear stochastic systems. IEEE Trans. Autom. Control 62, 1 (2016). MathSciNetGoogle Scholar
  17. 17.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hafayed, M., Abbas, S.: On near-optimal mean-field stochastic singular controls: necessary and sufficcient conditions for near-optimality. J. Optim. Theory Appl. 160(3), 778–808 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hafayed, M., Abbas, A., Abba, S.: On mean-field stochastic maximum principle for near-optimal controls for Poisson jump diffusion with applications. Int. J. Dyn. Control 2, 262–284 (2014)CrossRefGoogle Scholar
  20. 20.
    Hafayed, M., Abbas, S.: Stochastic near-optimal singular controls for jump diffusions: necessary and sufficient conditions. J. Dyn. Control Syst. 19(4), 503–517 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hafayed, M., Abbas, S., Veverka, P.: On necessary and sufficient conditions for near-optimal singular stochastic controls. Optim. Lett. 7(5), 949–966 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hafayed, M., Veverka, P., Abbas, S.: On maximum principle of near-optimality for diffusions with Jumps, with Application to consumption–investment problem. Differ. Equ. Dyn. Syst. 20(2), 111–125 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hafayed, M., Veverka, P., Abbas, A.: On near-optimal necessary and sufficient conditions for forward–backward stochastic systems with jumps, with applications to finance. Appl. Math. 59(4), 407–440 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Huang, J.H., Li, X., Wang, G.C.: Near-optimal control problems for linear forward–backward stochastic systems. Automatica 46, 397–404 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hui, E., Huang, J.H., Li, X., Wang, G.C.: Near-optimal control for stochasic recursive problems. Syst. Control Lett. 60, 161–168 (2011)CrossRefzbMATHGoogle Scholar
  26. 26.
    Zhang, L.Q., Huang, J.H., Li, X.: Necessary condition for near optimal control of linear forward–backward stochastic differential equations. Int. J. Control 88(8), 1594–1608 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zhang, L.Q.: Sufficient condition for near-optimal control of general controlled linear forward–backward stochastic differential equations. Int. J. Dyn. Control 5(2), 306–313 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Tang, M.N.: Stochastic maximum principle of near-optimal control of fully coupled forward–backward stochastic differential equation. In: Abstract and Applied Analysis 2014, 12 (2014).
  29. 29.
    Nie, T.Y., Shi, J.T., Wu Z.: Connection between MP and DPP for stochastic recursive optimal control problems: viscosity solution framework in local case. In: Proceedings of 2016 American Control Conference, pp. 7225–7230, July 6–8, Boston (2016)Google Scholar
  30. 30.
    Nie, T.Y., Shi, J.T., Wu, Z.: Connection between MP and DPP for stochastic recursive optimal control problems: viscosity solution framework in general case. SIAM J. Control Optim. 55(5), 3258–3294 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Hu, Y., Peng, S.G.: Solution of forward–backward stochastic differential equations. Probab. Theory Relat. Fields 103, 273–283 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Yong, J.M., Zhou, X.Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  33. 33.
    Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Peng, S.G.: A general stochastic maximum principle for optimal control problem. SIAM J. Control Optim. 28, 966–979 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Yong, J.M.: Optimality variational principle for controlled forward–backward stochastic differential equations with mixed initial–terminal conditions. SIAM J. Control Optim. 48, 4119–4156 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Wu, Z.: A general maximum principle for optimal control of forward–backward stochastic systems. Automatica 49, 1473–1480 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Hu, M.S.: Stochastic global maximum principle for optimization with recursive utilities. Probab. Uncertain. Quant. Risk 2, 1 (2017)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Zhang, L.Q.: Stochastic verification theorem of forward–backward controlled systems for viscosity solutions. Syst. Control Lett. 61, 649–654 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of ScienceBeijing University of Posts and TelecommunicationsBeijingChina

Personalised recommendations