基于路径依赖PDE粘性解理论的路径依赖的随机控制与对策问题

Nowadays path-dependent stochastic optimal control problems and differential games are hot issues in control theory, there are applied extensively in engineering and finance practice...This proposal is mainly concerned in studying the path-dependent stochastic control problem and differential game of general form by establishing the theory of viscosity solution for path-dependent fully nonlinear second order parabolic partial differential equations. We will choose sequences of suitable locally compact subspaces of the path space on which we define the jet function, which is the key technique to put forward a novel notion of viscosity solution, and provides a powerful analysis tool to handle path-dependent control problems. By dynamic programming principle and other tools of control theory, we are going to prove that the unique viscosity solution of Path-dependent Bellman equation characterizes the value functional of the corresponding optimal control problem. We will show the relationship between value functionals of path-dependent differential games and corresponding path-dependent Isaacs equations. By the uniqueness of the viscosity solution it will show that the upper and lower value functionals are coincide under Isaacs condition...We will apply the theory to some practical and special path-dependent models including time-delaying control problems and the pricing of look-back options, wish to provide some explicit and numerical solutions.

路径依赖的随机最优控制和微分对策问题是控制理论中的热点问题，在工程和金融界都有大量的应用背景。..本项目旨在研究一般形式的路径依赖的随机最优控制和微分对策问题。为此我们将首先建立路径依赖的完全非线性二阶抛物型偏微分方程理论，通过选择适当的具有局部紧性的路径空间的子空间来定义jet泛函，由此得到全新的路径空间粘性解定义，为解决路径依赖的控制问题提供有力的分析工具；其次通过动态规划原理证明和其他控制论技术证明路径依赖Bellman方程的唯一粘性解刻画了相应的路径依赖随机最优控制问题；另外建立路径依赖随机微分对策的值泛函与相应路径依赖Isaacs方程粘性解联系，并由粘性解的唯一性得到在Isaacs条件下对策问题的上下值泛函相等。..我们拟将所建立的理论应用到实际模型中，如时滞系统控制、回望期权定价等特殊路径依赖的控制问题，期望利用上述工具为这些问题提供显式解和数值解。

本项目研究了路径依赖框架下（非马尔可夫型）的随机系统的最优控制和博弈问题，以及与随机系统相关的(随机)PDE的正则性理论。取得了以下成果：1.研究非马尔可夫型跳过程和布朗运动共同驱动的一般形式线性系统的二次最优控制问题，利用动态规划和随机逆流技术得到与之关联的跳过程驱动的倒向随机Riccati方程可料解的存在唯一性。2.研究各分支相关联的高维平方根型退化随机系统的遍历理论，利用击中时估计，得到了与此过程相关联的PDE解的Holder估计。3.研究随机抛物型偏微分系统可解性理论，提出了一个一般的条件，并在此条件下得到了抛物系统的随机型Shauder估计，并得到了古典解的存在唯一性。4.研究了跳过程驱动的部分观测的倒向随机系统的博弈问题，得到了纳什均衡解的最大值原理刻画。成果相关文章发表在SIAM J. Control & Optimization, Stoch. Proc. Appl. 和 Ann. Inst. H Poincare, Prob. Stat.等杂志。