| 张伏,唐矛宁,孟庆欣.L\'evy过程驱动的正倒向随机系统的随机最大值原理[J].数学年刊A辑,2014,35(1):83~100 |
| L\'evy过程驱动的正倒向随机系统的随机最大值原理 |
| Stochastic Maximum Principle for Forward-Backward Stochastic Systems Associatedwith L′evy Processes |
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| DOI： |
| 中文关键词: 随机控制, 随机最大值原理, L\'{e}vy 过程, Teugels鞅, 正倒向随机微分方程 |
| 英文关键词:Stochastic control, Stochastic maximum principle,
L\'{e}vy processes, Teugels martingales, Forward-backward stochastic
differential equations |
| 基金项目:国家自然科学基金 (No.11101140, No.11301177, No.10325101, No.11171076), 中国博士后基金(No.2011M500721, No.2012T50391)和浙江省自然科学基金 (No.Y6110775, No.Y6110789) |
| Author Name | Affiliation | E-mail | | ZHANG Fu | Department of Mathematical Sciences, Huzhou University, Huzhou 313000, Zhejiang, China. | 09110180028@fudan.edu.cn | | TANG Maoning | Department of Mathematical Sciences, Huzhou University, Huzhou 313000, Zhejiang, China. | tmorning@hutc.zj.cn | | MENG Qingxin | Department of Mathematical Sciences, Huzhou University, Huzhou 313000, Zhejiang, China. | 071018034@fudan.edu.c |
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| 中文摘要: |
| 研究了由Teugels鞅和与之独立的多维Brown运动共同驱动的正倒向随机控制系统的最优控制问题.
这里Teugels鞅是一列与L\'{e}vy 过程相关的两两强正交的正态鞅 (见Nualart, Schoutens 在2000年的结果).
在允许控制值域为一非空凸闭集假设下, 采用凸变分法和对偶技术获得了最优控制存在所满足的充分和必要条件.
作为应用, 系统研究了线性正倒向随机系统的二次最优控制问题(简记为FBLQ问题), 通过相应的随机哈密顿系统对最优控制
进行了对偶刻画. 这里的随机哈密顿系统是由Teugels鞅和多维Brown运动共同驱动的线性正倒向随机微分方程,
其由状态方程、伴随方程和最优控制的对偶表示共同来构成. |
| 英文摘要: |
| The paper is concerned with a stochastic optimal control problem,
where the controlled systems are forward-backward stochastic
differential equations (or FBSDEs for short) driven by Teugels martingales and
an independent multi-dimensional Brownian motion. Here Teugels
martingales are a family of pairwise strongly orthonormal
martingales associated with L\'{e}vy processes (see the results of Nualart and Schoutens in 2000). We derive the necessary and sufficient
conditions for the existence of the optimal control in a stochastic
maximum principle by means of
convex variation methods and duality techniques when the control
domain is convex. As an application, the optimal control problem of
linear forward-backward stochastic systems with a quadratic cost
criteria (called forward-backward linear-quadratic problem, or FBLQ
problem for short) is discussed and characterized by stochastic
Hamilton system. Here, the stochastic Hamilton system is a
linear forward-backward stochastic differential equation with double
dimensions driven by Teugels martingales and an independent
multi-dimensional Brownian motion, consisting of state equation,
adjoint equation and dual presentation of the optimal control. |
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