Some results on pointwise second-order necessary conditions for stochastic optimal controls

• Published in: Science China. Mathematics Vol. 59; no. 2; pp. 227 - 238
• Main Authors: Zhang, HaiSen, Zhang, Xu
• Format: Journal Article
• Language: English
• Published: Beijing Science China Press 01.02.2016
• Subjects: Adjoints; Applications of Mathematics; Brief Reports; China; Diffusion; Drift; Mathematical analysis; Mathematics; Mathematics and Statistics; Maximum principle; Optimal control; Stochasticity; 一阶必要条件; 二阶; 奇异最优控制; 控制区域; 控制变量; 控制点; 最大值原理; 随机最优控制; stochastic optimal control; Pontryagin-type maximum principle; pointwise second-order necessary condition; 93E20; 60H10; needle variation; 60H07; Malliavin calculus
• ISSN: 1674-7283; 1869-1862
• DOI: 10.1007/s11425-015-5080-7

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.