Partially observed optimal controls of forward-backward doubly stochastic systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 828-843.

The partially observed optimal control problem is considered for forward-backward doubly stochastic systems with controls entering into the diffusion and the observation. The maximum principle is proven for the partially observable optimal control problems. A probabilistic approach is used, and the adjoint processes are characterized as solutions of related forward-backward doubly stochastic differential equations in finite-dimensional spaces. Then, our theoretical result is applied to study a partially-observed linear-quadratic optimal control problem for a fully coupled forward-backward doubly stochastic system.

DOI : 10.1051/cocv/2012035
Classification : 93E20, 60H10
Mots clés : forward-backward doubly stochastic system, partially observed optimal control, maximum principle, adjoint equation
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     title = {Partially observed optimal controls of forward-backward doubly stochastic systems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
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Shi, Yufeng; Zhu, Qingfeng. Partially observed optimal controls of forward-backward doubly stochastic systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 828-843. doi : 10.1051/cocv/2012035. http://archive.numdam.org/articles/10.1051/cocv/2012035/

[1] F. Baghery and B. Øksendal, A maximum principle for stochastic control with partial information. Stoch. Anal. Appl. 25 (2007) 705-717. | MR | Zbl

[2] S. Bahlali and B. Gherbal, Optimality conditions of controlled backward doubly stochastic differential equations. Random Oper. Stoch. Equ. 18 (2010) 247-265. | MR | Zbl

[3] A. Bally and A. Matoussi, Weak solutions of stochastic PDEs and backward doubly stochastic differential equations. J. Theoret. Probab. 14 (2001) 125-164. | MR | Zbl

[4] J.S. Baras, R.J. Elliott and M. Kohlmann, The partially-observed stochastic minimum principle. SIAM J. Control Optim. 27 (1989) 1279-1292. | MR | Zbl

[5] A. Bensoussan, Maximum principle and dynamic programming approaches of the optimal control of partially observed diffusions. Stochastics 9 (1983) 169-222. | MR | Zbl

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

[7] Y. Han, S. Peng and Z. Wu, Maximum principle for backward doubly stochastic control systems with applications. SIAM J. Control Optim. 48 (2010) 4224-4241. | MR | Zbl

[8] U.G. Haussmann, The maximum principle for optimal control of diffusions with partial information. SIAM J. Control Optim. 25 (1987) 341-361. | MR | Zbl

[9] L. Hu and Y. Ren, Stochastic PDIEs with nonlinear Neumann boundary conditions and generalized backward doubly stochastic differential equations driven by Lévy processes. J. Comput. Appl. Math. 229 (2009) 230-239. | MR | Zbl

[10] J. Huang, G. Wang and J. Xiong, A maximum principle for partial information backward stochastic control problems with applications. SIAM J. Control Optim. 48 (2009) 2106-2117. | MR | Zbl

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

[12] X. Li and S. Tang, General necessary conditions for partially-observed optimal stochastic controls. J. Appl. Probab. 32 (1995) 1118-1137. | MR | Zbl

[13] Q. Meng, A maximum principle for optimal control problem of fully coupled forward-backward stochastic systems with partial information. Sci. China Ser. A 52 (2009) 1579-1588. | MR | Zbl

[14] Q. Meng and M. Tang, Necessary and sufficient conditions for optimal control of stochastic systems associated with Lévy processes. Sci. China Ser. F 52 (2009) 1982-1992. | MR | Zbl

[15] E. Pardoux and S. Peng, Backward doubly stochastic differential equations and systems of quasilinear parabolic SPDE's. Probab. Theory Related Fields 98 (1994) 209-227. | MR | Zbl

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

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

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

[19] Y. Ren, A. Lin and L. Hu, Stochastic PDIEs and backward doubly stochastic differential equations driven by Lévy processes. J. Comput. Appl. Math. 223 (2009) 901-907. | MR | Zbl

[20] Y. Shi, Y. Gu and K. Liu, Comparison theorems of backward doubly stochastic differential equations and applications. Stoch. Anal. Appl. 23 (2005) 97-110. | MR | Zbl

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

[22] J. Shi and Z. Wu, Maximum principle for partially-observed optimal control of fully-coupled forward-backward stochastic systems. J. Optim. Theory Appl. 145 (2010) 543-578. | MR | Zbl

[23] S. Tang, The maximum principle for partially observed optimal control of stochastic differential equations. SIAM J. Control Optim. 36 (1998) 1596-1617. | MR | Zbl

[24] G. Wang and Z. Wu, The maximum principle for stochastic recursive optimal control problems under partial information. IEEE Trans. Autom. Control 54 (2009) 1230-1242. | MR

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

[26] Z. Wu, A maximum principle for partially observed optimal control of forward-backward stochastic control systems. Sci. China Ser. F 53 (2010) 2205-2214. | MR | Zbl

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

[28] L. Zhang and Y. Shi, Maximum principle for forward-backward doubly stochastic control systems and applications. ESAIM: COCV 17 (2011) 1174-1197. | Numdam | MR | Zbl

[29] L. Zhang and Y. Shi, Optimal Control of Stochastic Partial Differential Equations (2010) arXiv:1009.6061v2.

[30] Q. Zhang and H. Zhao, Pathwise stationary solutions of stochastic partial differential equations and backward doubly stochastic differential equations on infinite horizon. J. Funct. Anal. 252 (2007) 171-219. | MR | Zbl

[31] Q. Zhang and H. Zhao, Stationary solutions of SPDEs and infinite horizon BDSDEs under non-Lipschitz coefficients. J. Differ. Equations 248 (2010) 953-991. | MR | Zbl

[32] Q. Zhu, T. Wang and Y. Shi, Maximum principle for partially observed optimal control of backward doubly stochastic systems, in Proc. of the 30th Chinese Control Conference (2011) 1383-1388.

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