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

DOI : https://doi.org/10.1051/cocv/2010042
Classification : 93E20,  60H10
Mots clés : maximum principle, stochastic optimal control, forward-backward doubly stochastic differential equations, spike variations, variational equations, stochastic partial differential equations, nonzero sum stochastic differential game
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Zhang, Liangquan; Shi, Yufeng. Maximum principle for forward-backward doubly stochastic control systems and applications. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 4, pp. 1174-1197. doi : 10.1051/cocv/2010042. http://www.numdam.org/articles/10.1051/cocv/2010042/

[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 384185 | Zbl 0254.90066

[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 705930 | Zbl 0505.93078

[3] A. Bensoussan, Stochastic maximum principle for distributed parameter system. J. Franklin Inst. 315 (1983) 387-406. | MR 713370 | Zbl 0519.93042

[4] A. Bensoussan, Stochastic Control of Partially Observable Systems. Cambridge University Press, Cambridge (1992). | MR 1191160 | Zbl 0776.93094

[5] J.M. Bismut, An introductory approach to duality in optimal stochastic control. SIAM Rev. 20 (1978) 62-78. | MR 469466 | Zbl 0378.93049

[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 1626817 | Zbl 0916.93084

[7] T. Eisele, Nonexistence and nonuniqueness of open-loop equilibria in linear-quadratic differential games. J. Math. Anal. Appl. 37 (1982) 443-468. | MR 669840 | Zbl 0465.90099

[8] A. Friedman, Differential Games. Wiley-Interscience, New York (1971). | MR 421700 | Zbl 0229.90060

[9] S. Hamadène, Nonzero sum linear-quadratic stochastic differential games and backwad-forward equations. Stoch. Anal. Appl. 17 (1999) 117-130. | MR 1671515 | Zbl 0922.60050

[10] U.G. Haussmann, General necessary conditions for optimal control of stochastic systems. Math. Program. Stud. 6 (1976) 34-48. | MR 459883 | Zbl 0369.93048

[11] U.G. Haussmann, A stochastic maximum principle for optimal control of diffusions, Pitman Research Notes in Mathematics 151. Longman (1986). | MR 872471 | Zbl 0616.93076

[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 2346931 | Zbl 1132.93050

[13] H.J. Kushner, Necessary conditions for continuous parameter stochastic optimization problems. SIAM J. Control Optim. 10 (1972) 550-565. | MR 314535 | Zbl 0242.93063

[14] R.E. Mortensen, Stochastic optimal control with noisy observations. Int. J. Control 4 (1966) 455-464. | MR 213184 | Zbl 0201.48305

[15] M. Nisio, Optimal control for stochastic partial differential equations and viscosity solutions of Bellman equations. Nagoya Math. J. 123 (1991) 13-37. | MR 1126181 | Zbl 0749.93083

[16] D. Nualart and E. Pardoux, Stochastic calculus with anticipating integrands. Probab. Theory Relat. Fields 78 (1988) 535-581. | MR 950346 | Zbl 0629.60061

[17] B. Øksendal, Optimal control of stochastic partial differential equations. Stoch. Anal. Appl. 23 (2005) 165-179. | Zbl 1156.93406

[18] E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14 (1990) 55-61. | MR 1037747 | Zbl 0692.93064

[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 1258986 | Zbl 0792.60050

[20] S. Peng, A general stochastic maximum principle for optimal control problem. SIAM J. Control Optim. 28 (1990) 966-979. | MR 1051633 | Zbl 0712.93067

[21] S. Peng, Backward stochastic differential equations and application to optimal control. Appl. Math. Optim. 27 (1993) 125-144. | MR 1202528 | Zbl 0769.60054

[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 1989279 | Zbl 1031.60055

[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 1675098 | Zbl 0931.60048

[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 2230926

[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 1651258 | Zbl 0938.93066

[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 2136983 | Zbl 1156.93409

[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 1359179 | Zbl 0862.93067

[29] J. Yong and X.Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999). | MR 1696772 | Zbl 0943.93002

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