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Stochastic maximum principle for mixed regular-singular control problems of forward-backward systems

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Abstract

This paper considers a stochastic optimal control problem of a forward-backward system with regular-singular controls where the set of regular controls is not necessarily convex and the regular control enters the diffusion coefficient. This control problem is difficult to solve with the classical method of spike variation. The authors use the approach of relaxed controls to establish maximum principle for this stochastic optimal control problem. Sufficient optimality conditions are also investigated.

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References

  1. Bather J and Chernoff H, Sequential decision in the control of a spaceship (finite fuel), Journal of Applied Probability, 1967, 4(3): 584–604.

    Article  MathSciNet  MATH  Google Scholar 

  2. Benes V E, Shepp L A, and Witsenhausen H S, Some solvable stochastic control problems, Stoch. Stoch. Rep., 1980, 4: 39–83.

    MathSciNet  MATH  Google Scholar 

  3. Alvarez L H, Singular stochastic control, linear diffusions, and optimal stopping: A class of solvable problems, SIAM J. Control Optim., 2001, 39(6): 1697–1710.

    Article  MathSciNet  MATH  Google Scholar 

  4. Haussmann U G and Suo W L, Singular optimal stochastic controls I: Existence, SIAM J. Control Optim., 1995, 33(3): 916–936.

    Article  MathSciNet  MATH  Google Scholar 

  5. Haussmann U G and Suo W L, Singular optimal stochastic controls II: Dynamic programming, SIAM J. Control Optim., 1995, 33: 937–959.

    Article  MathSciNet  MATH  Google Scholar 

  6. Karatzas I and Shreve S E, Connections between optimal stopping and singular stochastic control I: Monotone follower problems, SIAM J. Control Optim., 1984, 22(6): 856–877.

    Article  MathSciNet  MATH  Google Scholar 

  7. Karatzas I and Shreve S E, Connections between optimal stopping and singular stochastic control II: Reflected follower problems, SIAM J. Control Optim., 1985, 23(3): 433–451.

    Article  MathSciNet  MATH  Google Scholar 

  8. Peng S G, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 1990, 28(4): 966–979.

    Article  MathSciNet  MATH  Google Scholar 

  9. Peng S G, Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim., 1993, 27(2): 125–144.

    Article  MathSciNet  MATH  Google Scholar 

  10. Shi J T and Wu Z, The maximum principle for fully coupled forward-backward stochastic control systems, Acta Automatica Sinica, 2006, 32(2): 161–169.

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Wu Z, Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems, Systems Sci. Math. Sci., 1998, 11: 249–259.

    MathSciNet  MATH  Google Scholar 

  13. Xu W S, Stochastic maximum principle for optimal control problem of forward and backward system, J. Austral. Math. Soc. Ser. B, 1995, 37(2): 172–185.

    Article  MathSciNet  MATH  Google Scholar 

  14. Bahlali S, Necessary and sufficient optimality conditions for relaxed and strict control problems, SIAM J. Control Optim., 2008, 47(4): 2078–2095.

    Article  MathSciNet  MATH  Google Scholar 

  15. Wu Z, A general stochastic maximum principle for optimal control problems of forward-backward stochastic control systems, Automatica, 2013, 49, DOI: 10.1016/j.automatica.2013.02.005.

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Bahlali S, A general stochastic maximum principle for optimal control problems of forward-backward systems, arXiv: 0801.4326v6 [math.PR].

  18. Cadenillas A and Haussmann U G, The stochastic maximum principle for a singular control problem, Stochastics: An International Journal of Probability and Stochastic Processes, 1994, 49(3–4): 211–237.

    MathSciNet  MATH  Google Scholar 

  19. Bahlali S and Chala A, The stochastic maximum principle in optimal control of singular diffusions with nonlinear coefficients, Random Oper. and Stoch. Equ., 2005, 13(1): 1–10.

    Article  MathSciNet  MATH  Google Scholar 

  20. Bahlali S and Mezerdi B, A general stochastic maximum principle for singular control problems, Electron J. Probab., 2005, 10: 988–1004.

    MathSciNet  Google Scholar 

  21. El Karoui N, Huu Nguyen N, and Jeanblanc Piqué M, Compactification methods in the control of degenerate diffusions: Existence of an optimal control, Stochastics, 1987, 20(3): 169–219.

    Article  MathSciNet  MATH  Google Scholar 

  22. Lou H W, Analysis of the optimal relaxed control to an optimal control problem, Appl. Math. Optim., 2009, 59(1): 75–97.

    Article  MathSciNet  MATH  Google Scholar 

  23. Yong J M and Zhou X Y, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.

    Book  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, 250014, China

    Feng Zhang

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  1. Feng Zhang
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Correspondence to Feng Zhang.

Additional information

This research was supported by the National Natural Science Foundation of China under Grant Nos. 11201268 and 61105077, and the Natural Science Foundation of Shandong Province under Grant Nos. ZR2011AQ018 and ZR2012AQ013.

This paper was recommended for publication by Editor ZHANG Jifeng.

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Zhang, F. Stochastic maximum principle for mixed regular-singular control problems of forward-backward systems. J Syst Sci Complex 26, 886–901 (2013). https://doi.org/10.1007/s11424-013-0287-6

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  • DOI: https://doi.org/10.1007/s11424-013-0287-6

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