Maximum principle for forward-backward doubly stochastic control systems and applications

Zhang, Liangquan; Shi, Yufeng

ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 4, pp. 1174-1197.

The maximum principle for optimal control problems of fully coupled forward-backward doubly stochastic differential equations (FBDSDEs in short) in the global form is obtained, under the assumptions that the diffusion coefficients do not contain the control variable, but the control domain need not to be convex. We apply our stochastic maximum principle (SMP in short) to investigate the optimal control problems of a class of stochastic partial differential equations (SPDEs in short). And as an example of the SMP, we solve a kind of forward-backward doubly stochastic linear quadratic optimal control problems as well. In the last section, we use the solution of FBDSDEs to get the explicit form of the optimal control for linear quadratic stochastic optimal control problem and open-loop Nash equilibrium point for nonzero sum stochastic differential games problem.

[1] A. Bensoussan, Point de Nash dans le cas de fonctionnelles quadratiques et jeux différentiels à N personnes. SIAM J. Control 12 (1974) 460-499. | MR | Zbl

[2] A. Bensoussan, Lectures on stochastic control, in Nonlinear Filtering and Stochastic Control, S.K. Mitter and A. Moro Eds., Lecture Notes in Mathematics 972, Springer-verlag, Berlin (1982). | MR | Zbl

[3] A. Bensoussan, Stochastic maximum principle for distributed parameter system. J. Franklin Inst. 315 (1983) 387-406. | MR | Zbl

[4] A. Bensoussan, Stochastic Control of Partially Observable Systems. Cambridge University Press, Cambridge (1992). | MR | Zbl

[5] J.M. Bismut, An introductory approach to duality in optimal stochastic control. SIAM Rev. 20 (1978) 62-78. | MR | Zbl

[6] S. Chen, X. Li and X. Zhou, Stochstic linear quadratic regulators with indefinite control weight cost. SIAM J. Control Optim. 36 (1998) 1685-1702. | MR | Zbl

[7] T. Eisele, Nonexistence and nonuniqueness of open-loop equilibria in linear-quadratic differential games. J. Math. Anal. Appl. 37 (1982) 443-468. | MR | Zbl

[8] A. Friedman, Differential Games. Wiley-Interscience, New York (1971). | MR | Zbl

[9] S. Hamadène, Nonzero sum linear-quadratic stochastic differential games and backwad-forward equations. Stoch. Anal. Appl. 17 (1999) 117-130. | MR | Zbl

[10] U.G. Haussmann, General necessary conditions for optimal control of stochastic systems. Math. Program. Stud. 6 (1976) 34-48. | MR | Zbl

[11] U.G. Haussmann, A stochastic maximum principle for optimal control of diffusions, Pitman Research Notes in Mathematics 151. Longman (1986). | MR | Zbl

[12] S. Ji and X.Y. Zhou, A maximum principle for stochastic optimal control with terminal state constraints, and its applications. Commun. Inf. Syst. 6 (2006) 321-338. | MR | Zbl

[13] H.J. Kushner, Necessary conditions for continuous parameter stochastic optimization problems. SIAM J. Control Optim. 10 (1972) 550-565. | MR | Zbl

[14] R.E. Mortensen, Stochastic optimal control with noisy observations. Int. J. Control 4 (1966) 455-464. | MR | Zbl

[15] M. Nisio, Optimal control for stochastic partial differential equations and viscosity solutions of Bellman equations. Nagoya Math. J. 123 (1991) 13-37. | MR | Zbl

[16] D. Nualart and E. Pardoux, Stochastic calculus with anticipating integrands. Probab. Theory Relat. Fields 78 (1988) 535-581. | MR | Zbl

[17] B. Øksendal, Optimal control of stochastic partial differential equations. Stoch. Anal. Appl. 23 (2005) 165-179. | Zbl

[18] E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14 (1990) 55-61. | MR | Zbl

[19] E. Pardoux and S. Peng, Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probab. Theory Relat. Fields 98 (1994) 209-227. | MR | Zbl

[20] S. Peng, A general stochastic maximum principle for optimal control problem. SIAM J. Control Optim. 28 (1990) 966-979. | MR | Zbl

[21] S. Peng, Backward stochastic differential equations and application to optimal control. Appl. Math. Optim. 27 (1993) 125-144. | MR | Zbl

[22] S. Peng and Y. Shi, A type of time-symmetric forward-backward stochastic differential equations. C. R. Acad. Sci. Paris, Sér. I 336 (2003) 773-778. | MR | Zbl

[23] S. Peng and Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J. Control Optim. 37 (1999) 825-843. | MR | Zbl

[24] L.S. Pontryagin, V.G. Boltyanskti, R.V. Gamkrelidze and E.F. Mischenko, The Mathematical Theory of Optimal Control Processes. Interscience, John Wiley, New York (1962).

[25] J. Shi and Z. Wu, The maximum principle for fully coupled forward-backward stochastic control system. Acta Automatica Sinica 32 (2006) 161-169. | MR

[26] Z. Wu, Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems. Systems Sci. Math. Sci. 11 (1998) 249-259. | MR | Zbl

[27] Z. Wu, Forward-backward stochastic differential equation linear quadratic stochastic optimal control and nonzero sum differential games. Journal of Systems Science and Complexity 18 (2005) 179-192. | MR | Zbl

[28] W. Xu, Stochastic maximum principle for optimal control problem of forward and backward system. J. Austral. Math. Soc. B 37 (1995) 172-185. | MR | Zbl

[29] J. Yong and X.Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999). | MR | Zbl