关于带递归效用的平均场最优控制问题的随机最大值原理

发布时间：2018-02-24 18:39

本文关键词： 递归效用函数平均场正倒向随机微分方程 Ekeland变分原理完全耦合的平均场正倒向随机微分方程　出处：《山东大学》2017年硕士论文　论文类型：学位论文

【摘要】：Peng[1]在1998年提出一个重要的公开问题:"除了一些特殊情形,当f非线性依赖于z时相应的全局最大值原理是一个公开问题"。[2]和[3]研究了这个公开问题,但是他们所得到的最大值原理仍包含未知参数。公开问题的主要难点在于倒向随机微分方程(简记为BSDE)的生成元f(x,y,z,u)是非线性依赖于z的,这是无论在理论还是实际生活中都存在的一种典型状态。2015年Hu[4]最终解决了这个历时已久的公开问题。在Hu[4]的基础上,本文将结论推广至平均场情形,并考虑了完全耦合的部分可观测的平均场随机最优控制问题。本文主要分为两部分。第一部分,我们研究带递归效用函数的平均场最优控制问题的随机最大值原理。我们得到了带递归效用函数的平均场倒向随机微分方程(简记为MFBSDE)的变分方程,且得到了新的最大值原理。控制域不必为凸且MFBSDE的生成元可包含z。我们考虑如下状态方程:带递归效用的平均场随机最优控制问题为在可容许控制集μ[0,T]上最小化(2)式中的代价泛函J(u(·)),即找到最优控制u,使得我们的目的是得到最小化代价泛函时,最优控制u(·)所能满足的条件。我们的主要思想是利用Ekeland变分原理以及相应的伴随方程得到最优控制的必要条件。本部分的主要难点在于:(1)如何得到MFBSDE的二阶变分方程的具体形式(与[15]不同)。(2)关于z的变分方程的二次形式导致二阶伴随方程十分复杂。第二部分我们考虑了一种完全耦合的正倒向随机系统的部分可观测的平均场随机最优控制问题。在正向扩散系数不包含控制变量且控制域不必为凸的假设下,由Ekeland变分原理并利用针状变分法,我们得到庞特里亚金型的最大值原理,并且相关的伴随过程为平均场情形下的正倒向随机微分方程(简记为FBSDE)的解。而完全耦合的正倒向随机最优控制问题的一般最大值原理仍是一个公开问题。我们定义如下代价泛函:当系数满足Lipschitz条件、可积性条件和G-单调性条件时,方程(5)存在唯一解,其中正向扩散系数不包含控制。我们应用Ekeland变分原理及相应的伴随方程得到完全耦合情形下的部分可观测的平均场随机最大值原理。
[Abstract]:In 1998, Peng [1] put forward an important open problem: "except for some special cases, the corresponding global maximum principle when f nonlinear depends on z is an open problem." [2] and [3]. But their maximum principle still contains unknown parameters. The main difficulty of the open problem is that the generator of backward stochastic differential equation (BSDE) is nonlinear dependent on z. In 2015, Hu [4] finally solved this long-lasting open problem. On the basis of Hu [4], the conclusion is extended to the case of mean field. The stochastic optimal control problem of partially observable mean field is considered. This paper is divided into two parts. In this paper, we study the stochastic maximum principle for the mean field optimal control problem with recursive utility function. We obtain the variational equations of the mean field backward stochastic differential equation (MFBSDE) with recursive utility function. A new maximum principle is obtained. The control domain need not be convex and the generator of MFBSDE can contain z. we consider the following equation of state: the stochastic optimal control problem of mean field with recursive utility is minimized on the admissible control set 渭 [0T]. When we find the optimal control u, our aim is to obtain the minimized cost functional. Our main idea is to obtain the necessary condition of optimal control by using the Ekeland variational principle and the corresponding adjoint equation. The main difficulty of this part is how to get the second order variation of MFBSDE. The quadratic form of the variational equation for z (different from [15]) makes the second order adjoint equation very complicated. In the second part, we consider the partially observable plane of a fully coupled forward backward stochastic system. Under the assumption that the forward diffusion coefficient does not contain control variables and the control domain does not have to be convex, By using the Ekeland variational principle and the needle-like variational method, we obtain the maximum value principle of the Ponterria-gold type. And the associated adjoint process is the solution of forward backward stochastic differential equation (FBSDE) in the case of mean field. The general maximum principle of fully coupled forward backward stochastic optimal control problem is still an open problem. The following cost functional is defined: when the coefficient satisfies the Lipschitz condition, For the integrability condition and G-monotonicity condition, the equation has a unique solution. The forward diffusion coefficient does not contain control. By using the Ekeland variational principle and the corresponding adjoint equations we obtain the partially observable stochastic maximum principle of the mean field in the case of complete coupling.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O232

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