完全耦合的正倒向随机微分方程系统在非Lipschitz代价泛函下最优控制的存在性.docVIP

完全耦合的正倒向随机微分方程系统在非 Lipschitz 代价泛函下最优控制的存在性 摘要：运用凸分析中的最优存在定理，本文研究了最优随机控制的存在性。而所研究的随机系 统是完全耦合的线性正倒向随机微分方程，且其代价泛函是非 Lipschitz 的函数。一些典型的 模型，例如 LQ 问题，可以包含在本文所研究的随机系统的框架内。 关键词：变分方法和最优控制，最优控制存在性，正倒向随机微分方程，非 Lipschitz 代价泛 函，线性二次最优控制问题 中图分类号： O211 The Existence of Optimal Control for Fully Coupled Forward-Backward Stochastic Di?erential Equation System with Non-Lipschitz Cost Functional MENG Qingxin1, ZHANG Qi2 1 Department of Mathematical Sciences, Huzhou University, Huzhou 313000 2 School of Mathematical Sciences, Fudan University, Shanghai 200433 Abstract: In this paper, we study the existence of stochastic optimal control by a new application of the existence theorem of convex optimality. The stochastic system in our concern is a linear fully coupled forward-backward stochastic di?erential equation with a non-Lipschitz cost functional. Some typical examples, such as LQ problem, can be included in this studied stochastic system. Key words: calculus of variations and optimal control, existence of optimal control, forward-backward stochastic di?erential equations, non-Lipschitz cost functional, linear-quadratic problem -1- 0 Introduction Since the solvability of nonlinear backward stochastic di?erential equation (BSDE for short) was proved by Pardoux and Peng [1] in 1990, BSDE has a broad applications in many ?elds, such as stochastic control, mathematical ?nance, stochastic games, etc. Among these applications, BSDEs themselves as the stochastic control systems and their optimal control problems attracted a great many concerns of researchers. There has been a large amount of literature to study this topic, and most works were concerned with the stochastic maximum principle and the related problems. To name but a few, we recommend [2, 3, 4, 5, 6] to the reader. However, only few paper gave a discussion on the existence of optimal control for BSDE system. Once the optimal control exists the stochastic system can reach its optimal state. Needless to say, the existe