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Partially observed optimal controls of forward-backward doubly stochastic systems∗

Published online by Cambridge University Press:  03 June 2013

Yufeng Shi  and
Qingfeng Zhu
Yufeng Shi
Affiliation:
School of Mathematics, Shandong University, 250100 Jinan, P.R. China. [email protected]
Qingfeng Zhu
Affiliation:
School of Mathematics, Shandong University, 250100 Jinan, P.R. China. [email protected] School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, 250014 Jinan, P.R. China
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Abstract

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.

Keywords

Forward-backward doubly stochastic systempartially observed optimal controlmaximum principleadjoint equation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013 , pp. 828 - 843
Copyright
© EDP Sciences, SMAI, 2013

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References

Baghery, F. and Øksendal, B., A maximum principle for stochastic control with partial information. Stoch. Anal. Appl. 25 (2007) 705–717. Google Scholar
Bahlali, S. and Gherbal, B., Optimality conditions of controlled backward doubly stochastic differential equations. Random Oper. Stoch. Equ. 18 (2010) 247–265. Google Scholar
Bally, A. and Matoussi, A., Weak solutions of stochastic PDEs and backward doubly stochastic differential equations. J. Theoret. Probab. 14 (2001) 125–164. Google Scholar
Baras, J.S., Elliott, R.J. and Kohlmann, M., The partially-observed stochastic minimum principle. SIAM J. Control Optim. 27 (1989) 1279–1292. Google Scholar
Bensoussan, A., Maximum principle and dynamic programming approaches of the optimal control of partially observed diffusions. Stochastics 9 (1983) 169–222. Google Scholar
Bismut, J.M., An introductory approach to duality in optimal stochastic control. SIAM J. Control Optim. 20 (1978) 62–78. Google Scholar
Han, Y., Peng, S. and Wu, Z., Maximum principle for backward doubly stochastic control systems with applications. SIAM J. Control Optim. 48 (2010) 4224–4241. Google Scholar
Haussmann, U.G., The maximum principle for optimal control of diffusions with partial information. SIAM J. Control Optim. 25 (1987) 341–361. Google Scholar
Hu, L. and Ren, Y., 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. Google Scholar
Huang, J., Wang, G. and Xiong, J., A maximum principle for partial information backward stochastic control problems with applications. SIAM J. Control Optim. 48 (2009) 2106–2117. Google Scholar
Kushner, H.J., Necessary conditions for continuous parameter stochastic optimization problems. SIAM J. Control Optim. 10 (1972) 550–565. Google Scholar
Li, X. and Tang, S., General necessary conditions for partially-observed optimal stochastic controls. J. Appl. Probab. 32 (1995) 1118–1137. Google Scholar
Meng, Q., 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. Google Scholar
Meng, Q. and Tang, M., Necessary and sufficient conditions for optimal control of stochastic systems associated with Lévy processes. Sci. China Ser. F 52 (2009) 1982–1992. Google Scholar
Pardoux, E. and Peng, S., Backward doubly stochastic differential equations and systems of quasilinear parabolic SPDE’s. Probab. Theory Related Fields 98 (1994) 209–227. Google Scholar
Peng, S., A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28 (1990) 966–979. Google Scholar
Peng, S. and Shi, Y., A type of time-symmetric forward-backward stochastic differential equations. C. R. Acad. Sci. Paris, Ser. I 336 (2003) 773–778. Google Scholar
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).
Ren, Y., Lin, A. and Hu, L., Stochastic PDIEs and backward doubly stochastic differential equations driven by Lévy processes. J. Comput. Appl. Math. 223 (2009) 901–907. Google Scholar
Shi, Y., Gu, Y. and Liu, K., Comparison theorems of backward doubly stochastic differential equations and applications. Stoch. Anal. Appl. 23 (2005) 97–110. Google Scholar
Shi, J. and Wu, Z., The maximum principle for fully coupled forward backward stochastic control system. Acta Automat. Sinica 32 (2006) 161–169. Google Scholar
Shi, J. and Wu, Z., Maximum principle for partially-observed optimal control of fully-coupled forward-backward stochastic systems. J. Optim. Theory Appl. 145 (2010) 543–578. Google Scholar
Tang, S., The maximum principle for partially observed optimal control of stochastic differential equations. SIAM J. Control Optim. 36 (1998) 1596–1617. Google Scholar
Wang, G. and Wu, Z., The maximum principle for stochastic recursive optimal control problems under partial information. IEEE Trans. Autom. Control 54 (2009) 1230–1242. Google Scholar
Wu, Z., Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems. Syst. Sci. Math. Sci. 11 (1998) 249–259. Google Scholar
Wu, Z., A maximum principle for partially observed optimal control of forward-backward stochastic control systems. Sci. China Ser. F 53 (2010) 2205–2214. Google Scholar
Xu, W., Stochastic maximum principle for optimal control problem of forward and backward system. J. Aust. Math. Soc. 37 (1995) 172–185. Google Scholar
Zhang, L. and Shi, Y., Maximum principle for forward-backward doubly stochastic control systems and applications. ESAIM: COCV 17 (2011) 1174–1197. Google Scholar
L. Zhang and Y. Shi, Optimal Control of Stochastic Partial Differential Equations (2010) arXiv:1009.6061v2.
Zhang, Q. and Zhao, H., Pathwise stationary solutions of stochastic partial differential equations and backward doubly stochastic differential equations on infinite horizon. J. Funct. Anal. 252 (2007) 171–219. Google Scholar
Zhang, Q. and Zhao, H., Stationary solutions of SPDEs and infinite horizon BDSDEs under non-Lipschitz coefficients. J. Differ. Equations 248 (2010) 953–991. Google Scholar
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.