带跳跃平均场倒向随机微分方程的线性二次最优控制

数学物理学报 ›› 2019, Vol. 39 ›› Issue (3): 620-637.

带跳跃平均场倒向随机微分方程的线性二次最优控制

唐矛宁(),孟庆欣*()

湖州师范学院理学院浙江湖州 313000

收稿日期:2016-12-09 出版日期:2019-06-26 发布日期:2019-06-27
通讯作者: 孟庆欣 E-mail:tmorning@zjhu.edu.cn;mqx@zjhu.edu.cn
作者简介:唐矛宁, tmorning@zjhu.edu.cn
基金资助:
国家自然科学基金(11871121);浙江省自然科学基金杰出青年基金项目(LR15A010001)

Linear-Quadratic Optimal Control Problems for Mean-Field Backward Stochastic Differential Equations with Jumps

Maoning Tang(),Qingxin Meng*()

College of Science, Huzhou University, Zhejiang Huzhou 313000

Received:2016-12-09 Online:2019-06-26 Published:2019-06-27
Contact: Qingxin Meng E-mail:tmorning@zjhu.edu.cn;mqx@zjhu.edu.cn
Supported by:
the NSFC(11871121);the Natural Science Foundation of Zhejiang Province for Distinguished Young Scholar(LR15A010001)

摘要/Abstract

摘要：

该文研究了一类随机线性二次最优控制问题，其中状态方程是由泊松随机鞅测度和布朗运动共同驱动的平均场类型的倒向随机微分方程.首先，通过经典的凸变分原理获得了最优控制的存在性与唯一性；其次，利用对偶方法给出了最优控制的随机哈密顿系统刻画，这里的随机哈密顿系统是由状态方程、对偶方程和最优控制的对偶刻画构成的一个完全耦合的具有跳跃的平均场正倒向随机微分方程；最后，利用解耦技术，通过引入两个黎卡提方程和一个平均场倒向随机微分方程对随机哈密顿系统进行解耦，进而获得最优控制的反馈表示.

关键词: 平均场, 最优控制, 倒向随机微分方程, 对偶方程

Abstract:

This paper is concerned with a linear quadratic optimal control problem for meanfield backward stochastic differential equations driven by a Poisson random martingale measure and a Brownian motion. Firstly, by the classic convex variation principle, the existence and uniqueness of the optimal control is obtained. Secondly, the optimal control is characterized by the stochastic Hamilton system which turns out to be a linear fully coupled mean-field forward-backward stochastic differential equation with jumps by the duality method. Thirdly, in terms of a decoupling technique, the stochastic Hamilton system is decoupled by introducing two Riccati equations and a MF-BSDE with jumps. Then an explicit representation for the optimal control is obtained.

Key words: Mean-field, Optimal control, Backward stochastic Differential equation, Adjoint process

中图分类号:

O232

唐矛宁,孟庆欣. 带跳跃平均场倒向随机微分方程的线性二次最优控制[J]. 数学物理学报, 2019, 39(3): 620-637.

Maoning Tang,Qingxin Meng. Linear-Quadratic Optimal Control Problems for Mean-Field Backward Stochastic Differential Equations with Jumps[J]. Acta mathematica scientia,Series A, 2019, 39(3): 620-637.

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