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山东大学学报（理学版） ›› 2016, Vol. 51 ›› Issue (5): 121-129.doi: 10.6040/j.issn.1671-9352.0.2015.383

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完全耦合正倒向随机控制系统的动态规划原理与最大值原理之间的联系

聂天洋,史敬涛*   

  1. 山东大学数学学院, 山东 济南 250100
  • 收稿日期:2015-07-26 出版日期:2016-05-20 发布日期:2016-05-16
  • 通讯作者: 史敬涛(1978— ),男,博士,副教授,研究方向为随机控制、正倒向随机系统、时滞随机系统、金融数学.E-mail:shijingtao@sdu.edu.cn E-mail:nietianyang@sdu.edu.cn
  • 作者简介:聂天洋(1986— ),男,博士,助理研究员,研究方向为随机控制理论及其应用.E-mail: nietianyang@sdu.edu.cn
  • 基金资助:
    国家自然科学基金资助项目(61573217,11571205,11301011,11201264);山东省自然科学基金资助项目(JQ201401,ZR2015JL003);山东大学基本科研业务费资助项目(2015HW023)

The connection between DPP and MP for the fully coupled forward-backward stochastic control systems

NIE Tian-yang, SHI Jing-tao*   

  1. School of Mathematics, Shandong University, Jinan 250100, Shandong, China
  • Received:2015-07-26 Online:2016-05-20 Published:2016-05-16

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摘要： 研究了完全耦合正倒向随机控制系统的动态规划原理和最大值原理之间的联系,其递归效用泛函由受控完全耦合的正倒向随机微分方程的解给出。主要结果是在一定的光滑性假设下,给出了最优值函数、广义哈密顿函数和对偶过程之间的联系,但正向方程的扩散项不含变量z。一般情形的结果仍是公开问题。最后给出一个线性例子来解释理论结果。

关键词: 完全耦合正倒向随机微分方程, 最大值原理, 随机最优控制, 动态规划原理

Abstract: This paper is concerned with the connection between dynamic programming principle(DPP)and maximum principle(MP)for the forward-backward stochastic control system, where the recursive cost functional is defined as one of the solution to a controlled fully coupled forward-backward stochastic differential equation(FBSDE). With some smooth assumptions, relations among the value function, generalized Hamiltonian function and adjoint processes are given, when the diffusion coefficient of the forward equation does not contain the state variable z. The general case for the problem is open. A linear example is discussed as the illustration of our main result.

Key words: fully coupled forward-backward stochastic differential equation, maximum principle, stochastic optimal control, dynamic programming principle

中图分类号: 

  • O241.82
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