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Some results on pointwise second-order necessary conditions for stochastic optimal controls. (English) Zbl 1338.93409

Summary: The purpose of this paper is to derive some pointwise second-order necessary conditions for stochastic optimal controls in the general case that the control variable enters into both the drift and the diffusion terms. When the control region is convex, a pointwise second-order necessary condition for stochastic singular optimal controls in the classical sense is established; while when the control region is allowed to be nonconvex, we obtain a pointwise second-order necessary condition for stochastic singular optimal controls in the sense of Pontryagin-type maximum principle. It is found that, quite different from the first-order necessary conditions, the correction part of the solution to the second-order adjoint equation appears in the pointwise second-order necessary conditions whenever the diffusion term depends on the control variable, even if the control region is convex.

MSC:

93E20 Optimal stochastic control
60H07 Stochastic calculus of variations and the Malliavin calculus
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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References:

[1] Bensoussan, A., Lectures on stochastic control, 1-62 (1982), Berlin · Zbl 0505.93078
[2] Bismut J-M. An introductory approach to duality in optimal stochastic control. SIAM Rev, 1978, 20: 62-78 · Zbl 0378.93049
[3] Bonnans J F, Silva F J. First and second order necessary conditions for stochastic optimal control problems. Appl Math Optim, 2012, 65: 403-439 · Zbl 1244.49045
[4] Gabasov R, Kirillova F M. High order necessary conditions for optimality. SIAM J Control, 1972, 10: 127-168 · Zbl 0236.49005
[5] Haussmann U G. General necessary conditions for optimal control of stochastic systems. Math Prog Study, 1976, 6: 34-48 · Zbl 0369.93048
[6] Kushner H J. Necessary conditions for continuous parameter stochastic optimization problems. SIAM J Control Optim, 1972, 10: 550-565 · Zbl 0242.93063
[7] Nualart D. The Malliavin Calculus and Related Topics, 2nd ed. Berlin: Springer-Verlag, 2006 · Zbl 1099.60003
[8] Peng S. A general stochastic maximum principle for optimal control problems. SIAM J Control Optim, 1990, 28: 966-979 · Zbl 0712.93067
[9] Tang S. A second-order maximum principle for singular optimal stochastic controls. Discrete Contin Dyn Syst Ser B, 2010, 14: 1581-1599 · Zbl 1219.93147
[10] Yong J, Zhou X. Stochastic Controls: Hamiltonian Systems and HJB Equations. New York: Springer-Verlag, 1999 · Zbl 0943.93002
[11] Zhang H, Zhang X. Pointwise second-order necessary conditions for stochastic optimal controls, part I: The case of convex control constraint. SIAM J Control Optim, 2015, 53: 2267-2296 · Zbl 1337.49045
[12] Zhang H, Zhang X. Pointwise second-order necessary conditions for stochastic optimal controls, part II: The general case. ArXiv:1509.07995, 2015
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