Singular optimal controls for stochastic recursive systems under convex control constraint
Singular optimal controls for stochastic recursive systems under convex control constraint
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DOI：
10.1016/j.jmaa.2020.124905
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发表时间：
2021-05
影响因子：
1.3
通讯作者：
Liangquan Zhang
中科院分区：
文献类型：
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作者：
Liangquan Zhang
本文研究了两类由正倒向随机微分方程（简称FBSDEs）控制的奇异最优控制（简称SOC）问题，其中控制由正则控制和奇异控制两部分组成.漂移项和扩散项都可能涉及常规控制变量。正则控制域被假定为凸的。在一定的假设下，在Malliavin演算的框架下，我们得到了经典意义下随机SOC的逐点二阶必要条件。这个条件是由两个伴随过程，最大条件的哈密尔顿支持一个说明性的例子。得到了最优奇异控制的一个新的必要条件。此外，作为一个副产品，SOC的验证定理推导通过粘度解决方案，而不涉及任何导数的价值函数。值得指出的是，该定理比限制性经典验证定理具有更广泛的适用性。最后，在不假设值函数足够光滑的情况下，我们着重讨论了这类SOC问题的最大值原理和动态规划原理之间的联系。
In this paper, we study two kinds of singular optimal controls (SOCs for short) problems where the systems governed by forward-backward stochastic differential equations (FBSDEs for short), in which the control has two components: the regular control, and the singular one. Both drift and diffusion terms may involve the regular control variable. The regular control domain is postulated to be convex. Under certain assumptions, in the framework of the Malliavin calculus, we derive the pointwise second-order necessary conditions for stochastic SOC in the classical sense. This condition is described by two adjoint processes, a maximum condition on the Hamiltonian supported by an illustrative example. A new necessary condition for optimal singular control is obtained as well. Besides, as a by-product, a verification theorem for SOCs is derived via viscosity solutions without involving any derivatives of the value functions. It is worth pointing out that this theorem has wider applicability than the restrictive classical verification theorems. Finally, we focus on the connection between the maximum principle and the dynamic programming principle for such SOCs problem without the assumption that the value function is smooth enough.