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Stochastic controls with terminal contingent conditions. (English) Zbl 0937.93053

For an optimal control problem \[ J(u(\cdot))={\mathbf E}g(y(0))+{\mathbf E}\int_0^T\varphi(t, y(t), z(t), u(t)) dt\to\min , \] where \(dy(t)=f(t, y(t), z(t), u(t)) dt+z(t) dw(t),\) \(t<T\), \(y(T)=\xi\), and \(u(t)\) is a control vector, a necessary condition of optimality (maximum principle) is found and it is examined when it becomes sufficient.
In the linear-quadratic case of the control problem, the optimal control is obtained as a feedback from the solution to the forward-backward stochastic differential equation.
The study of the nonlinear optimal control problem with additional integral constraints employs the method proposed by N. Dokuchaev [Theor. Probab. Appl. 41, No. 4, 761-768 (1996; Zbl 0913.60038)].

MSC:

93E20 Optimal stochastic control
49N10 Linear-quadratic optimal control problems

Citations:

Zbl 0913.60038
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References:

[1] Arkin, V.; Saksonov, M. S., Necessary optimality conditions for stochastic differential equations, Soviet Math. Dokl., 20, 1-5 (1979) · Zbl 0414.49017
[2] Bensoussan, A., Lectures on stochastic control, part I, Lecture Notes in Math., 972, 1-39 (1983)
[3] Bismut, J. M., An introductory approach to duality in stochastic controls, SIAM Rev., 20, 62-78 (1978) · Zbl 0378.93049
[4] S. Chen, and, X. Y. Zhou, Stochastic linear quadratic regulators with indefinite control weight costs, II, preprint.; S. Chen, and, X. Y. Zhou, Stochastic linear quadratic regulators with indefinite control weight costs, II, preprint. · Zbl 1023.93072
[5] Dokuchaev, N. G., Optimal stopping of stochastic processes in a problem with constraints, Theory Probab. Appl., 41, 761-768 (1996) · Zbl 0913.60038
[6] Dynkin, E.; Evstigneev, I., Regular conditional expectations of correspondences, Theory Probab. Appl., 21, 325-338 (1976) · Zbl 0367.60002
[7] Fleming, W. H., Optimal control of partially observable diffusions, SIAM J. Control Optim., 6, 194-215 (1968) · Zbl 0167.09104
[8] Haussmann, U. G., A Stochastic Maximum Principle for Optimal Control of Diffusions (1986), Pitman: Pitman Boston/London/Melbourne · Zbl 0616.93076
[9] Karoui, N. El.; Peng, S.; Quenez, M. C., Backward stochastic differential equations in finance, Math. Finance, 7, 1-71 (1997) · Zbl 0884.90035
[10] Kushner, H. J., Necessity conditions for continuous parameter stochastic optimization problems, SIAM J. Control Optim., 10, 550-565 (1972) · Zbl 0242.93063
[11] Matveev, A. S.; Yakubovich, V. A., Nonconvex problems of the global optimization, St. Petersburg Math. J., 4, 1217-1243 (1993)
[12] Pardoux, E.; Peng, S., Adapted solutions of backward stochastic equations, Systems Control Lett., 14, 55-61 (1990) · Zbl 0692.93064
[13] Peng, S., A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28, 966-979 (1990) · Zbl 0712.93067
[14] Peng, S., Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim., 27, 125-144 (1993) · Zbl 0769.60054
[15] Yong, J.; Zhou, X. Y., Stochastic Controls: Hamiltonian Systems and HJB Equations (1999), Springer: Springer New York · Zbl 0943.93002
[16] Zhou, X. Y., A unified treatment of maximum principle and dynamic programming in stochastic controls, Stochastics Stochastics Rep., 36, 137-161 (1991) · Zbl 0756.93087
[17] Zhou, X. Y., Sufficient conditions of optimality for stochastic systems with controllable diffusions, IEEE Trans. Automat. Control, 41, 1176-1179 (1996) · Zbl 0857.93099
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