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Equivalent cost functionals and stochastic linear quadraticoptimal control problems

Published online by Cambridge University Press:  23 February 2012

Zhiyong Yu*
Affiliation:
School of Economics, Shandong University, Jinan 250100, P.R. China. [email protected]
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Abstract

This paper is concerned with the stochastic linear quadratic optimal control problems (LQproblems, for short) for which the coefficients are allowed to be random and the costfunctionals are allowed to have negative weights on the square of control variables. Wepropose a new method, the equivalent cost functional method, to deal with the LQ problems.Comparing to the classical methods, the new method is simple, flexible and non-abstract.The new method can also be applied to deal with nonlinear optimization problems.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2012

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References

B.D.O. Anderson and J.B. Moore, Optimal control-Linear quadratic methods. Prentice-Hall, New York (1989).
A. Bensoussan, Lecture on stochastic cntrol, Part I, in Nonlinear Filtering and Stochastic Control, Lecture Notes in Math. 972. Springer-Verlag, Berlin (1983) 1–39.
J.M. Bismut, Controle des systems linears quadratiques : applications de l’integrale stochastique, in Séminaire de Probabilités XII, Lecture Notes in Math. 649, edited by C. Dellacherie, P.A. Meyer and M. Weil. Springer-Verlag, Berlin (1978) 180–264.
Chen, S. and Yong, J., Stochastic linear quadratic optimal control problems. Appl. Math. Optim. 43 (2001) 2145. Google Scholar
Chen, S. and Zhou, Z., Stochastic linaer quadratic regulators with indefinite control weight costs. II. SIAM J. Control Optim. 39 (2000) 10651081. Google Scholar
Chen, S., Li, X. and Zhou, X., Stochastic linear quadratic regulators with indefinite control weight costs. SIAM J. Control Optim. 36 (1998) 16851702. Google Scholar
M.H.A. Davis, Linear estimation and stochastic control. Chapman and Hall, London (1977).
Hu, Y. and Peng, S., Solution of forward-backward stochastic differential equations. Prob. Theory Relat. Fields 103 (1995) 273283. Google Scholar
M. Jeanblanc and Z. Yu , Optimal investment problems with uncertain time horizon. Working paper.
Kalman, R.E., Contributions to the theory of optimal control. Bol. Soc. Math. Mexicana 5 (1960) 102119. Google Scholar
J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications, Lecture Notes in Math. 1702. Springer-Verlag, New York (1999).
S. Peng, New development in stochastic maximum principle and related backward stochastic differential equations, in proceedings of 31st CDC Conference. Tucson (1992).
S. Peng, Open problems on backward stochastic differential equations, in Control of Distributed Parameter and Stochastic Systems (Hangzhou, 1998). edited by S. Chen et al., Kluwer Academic Publishers, Boston (1999) 966–979.
Peng, S. and Wu, Z., Fully coupled forward-backward stochastic differential equation and applications to optimal control. SIAM J. Control Optim. 37 (1999) 825843. Google Scholar
R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, New Jersey (1970).
Tang, S., General linear quadratic optimal stochastic control problems with random coefficients : linear stochastic Hamilton systems and backward stochastic Riccati equations. SIAM J. Control Optim. 42 (2003) 5375. Google Scholar
Wonham, W.M., On a matrix Riccati equation of stochastic control. SIAM J. Control Optim. 6 (1968) 312326 . Google Scholar
Wu, Z., Forward-backward stochastic differential equations, linear quadratic stochastic optimal control and nonzero sum differential games. Journal of Systems Science and Complexity 18 (2005) 179192. Google Scholar
J. Yong and X. Zhou, Stochastic controls : Hamiltonian systems and HJB equations. Springer-Verlag, New York (1999).