Zhang, HaiSen; Zhang, Xu Some results on pointwise second-order necessary conditions for stochastic optimal controls. (English) Zbl 1338.93409 Sci. China, Math. 59, No. 2, 227-238 (2016). 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. Cited in 4 Documents MSC: 93E20 Optimal stochastic control 60H07 Stochastic calculus of variations and the Malliavin calculus 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) Keywords:stochastic optimal control; needle variation; Pontryagin-type maximum principle; pointwise second-order necessary condition; Malliavin calculus PDFBibTeX XMLCite \textit{H. Zhang} and \textit{X. Zhang}, Sci. China, Math. 59, No. 2, 227--238 (2016; Zbl 1338.93409) Full Text: DOI arXiv 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.