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Maximum principle for optimal control of fully coupledforward-backward stochastic differential delayed equations∗

Published online by Cambridge University Press:  16 January 2012

Jianhui Huang and
Jingtao Shi
Jianhui Huang
Affiliation:
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, P.R. China. majhuang@inet.polyu.edu.hk
Jingtao Shi
Affiliation:
School of Mathematics, Shandong University, Jinan 250100, P.R. China; shijingtao@sdu.edu.cn

Abstract

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This paper deals with the optimal control problem in which the controlled system isdescribed by a fully coupled anticipated forward-backward stochastic differential delayedequation. The maximum principle for this problem is obtained under the assumption that thediffusion coefficient does not contain the control variables and the control domain is notnecessarily convex. Both the necessary and sufficient conditions of optimality are proved.As illustrating examples, two kinds of linear quadratic control problems are discussed andboth optimal controls are derived explicitly.

Keywords

Stochastic optimal controlmaximum principlestochastic differential delayed equationanticipated backward differential equationfully coupled forward-backward stochastic systemClarke generalized gradient

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 4 , October 2012, pp. 1073 - 1096
Copyright
© EDP Sciences, SMAI, 2012

References

Antonelli, F., Backward-forward stochastic differential equations. Ann. Appl. Prob. 3 (1993) 777–793. Google Scholar
Antonelli, F., Baruccib, E. and Mancinoc, M.E., Asset pricing with a forward-backward stochastic differential utility. Econ. Lett. 72 (2001) 151–157. Google Scholar
Buckdahn, R. and Hu, Y., Hedging contingent claims for a large investor in an incomplete market. Adv. Appl. Prob. 30 (1998) 239–255. Google Scholar
L. Chen and Z. Wu, Maximum principle for stochastic optimal control problem of forward-backward system with delay, in Proc. Joint 48th IEEE CDC and 28th CCC, Shanghai, P.R. China (2009) 2899–2904.
Chen, L. and Wu, Z., Maximum principle for the stochastic optimal control problem with delay and application. Automatica 46 (2010) 1074–1080. Google Scholar
Chen, L. and Wu, Z., A type of generalized forward-backward stochastic differential equations and applications. Chin. Ann. Math. Ser. B 32 (2011) 279–292. Google Scholar
Cucoo, D. and Cvitanic, J., Optimal consumption choices for a ‘large’ investor. J. Econ. Dyn. Control 22 (1998) 401–436. Google Scholar
Cvitanic, J. and Ma, J., Hedging options for a large investor and forward-backward SDE’s. Ann. Appl. Prob. 6 (1996) 370–398. Google Scholar
J. Cvitanic, X.H. Wan and J.F. Zhang, Optimal contracts in continuous-time models. J. Appl. Math. Stoch. Anal. 2006 (2006), Article ID 95203 1–27.
El Karoui, N., Peng, S.G. and Quenez, M.C., Backward stochastic differential equations in finance. Math. Finance 7 (1997) 1–71. Google Scholar
Hu, Y. and Peng, S.G., Solution of forward-backward stochastic differential equations. Prob. Theory Relat. Fields 103 (1995) 273–283. Google Scholar
Kolmanovsky, V.B. and Maizenberg, T.L., Optimal control of stochastic systems with aftereffect, in Stochastic Systems, Translated from Avtomatika i Telemekhanika. 1 (1973) 47–61. Google Scholar
Meng, Q.X., A maximum principle for optimal control problem of fully coupled forward-backward stochastic systems with partial information. Sciences in China, Mathematics 52 (2009) 1579–1588. Google Scholar
S.E.A. Mohammed, Stochastic differential equations with memory : theory, examples and applications. Stochastic Analysis and Related Topics VI. The Geido Workshop, 1996. Progress in Probability, Birkhauser (1998).
B. Øksendal and A. Sulem, A maximum principle for optimal control of stochastic systems with delay, with applications to finance, Optimal Control and Partial Differential Equations – Innovations and Applications, edited by J.M. Menaldi, E. Rofman and A. Sulem. IOS Press, Amsterdam (2000).
Pardoux, E. and Peng, S.G., Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14 (1990) 55–61. Google Scholar
Peng, S.G., Backward stochastic differential equations and applications to optimal control. Appl. Math. Optim. 27 (1993) 125–144. Google Scholar
S.G. Peng, Backward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJB equations, Topics on Stochastic Analysis (in Chinese), edited by J.A. Yan, S.G. Peng, S.Z. Fang and L.M. Wu. Science Press, Beijing (1997) 85–138.
Peng, S.G. and Wu, Z., Fully coupled forward-backward stochastic differential equations and applications to the optimal control. SIAM J. Control Optim. 37 (1999) 825–843. Google Scholar
Peng, S.G. and Yang, Z., Anticipated backward stochastic differential equations. Ann. Prob. 37 (2009) 877–902. Google Scholar
Shi, J.T. and Wu, Z., The maximum principle for fully coupled forward-backward stochastic control systems. ACTA Automatica Sinica 32 (2006) 161–169. Google Scholar
Shi, J.T. and Wu, Z., The maximum principle for partially observed optimal control of fully coupled forward-backward stochastic system. J. Optim. Theory Appl. 145 (2010) 543–578. Google Scholar
Wang, G.C. and Wu, Z., The maximum principles for stochastic recursive optimal control problems under partial information. IEEE Trans. Autom. Control 54 (2009) 1230–1242. Google Scholar
N. Williams, On dynamic principal-agent problems in continuous time. Working paper (2008). Available on the website : http://www.ssc.wisc.edu/˜nwilliam/dynamic-pa1.pdf
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
Yong, J.M., Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions. SIAM J. Control Optim. 48 (2010) 4119–4156. Google Scholar
J.M. Yong and X.Y. Zhou, Stochastic Controls : Hamiltonian Systems and HJB Equations. Springer-Verlag, New York (1999).
Yu, Z.Y., Linear-quadratic optimal control and nonzero-sum differential game of forward-backward stochastic system. Asian J. Control 14 (2012) 1–13. Google Scholar
Zhang, J.F., The wellposedness of FBSDEs. Discrete Contin. Dyn. Syst., Ser. B 6 (2006) 927–940. Google Scholar
Zhou, X.Y., Sufficient conditions of optimality for stochastic systems with controllable diffusions. IEEE Trans. Autom. Control 41 (1996) 1176–1179. Google Scholar