随机控制系统中的滞后、超前问题研究及应用

We study the delayed and anticipated problems of the stochastic control systems and their applications in real world. In detail, by using the classical stochastic analysis theory and anticipated stochastic differential equation, we consider linear-quadratic optimal control problem for the systems described by stochastic functional delay equation, delayed non-zero sum stochastic differential game and their applications. Motivated by some problems arising in mathematical finance, such as stochastic differential utility with delay, we introduce a new type of systems described by backward delay stochastic differential equations. We want to study the maximum principle and linear-quadratic optimal problem of the backward controlled systems with delay using the classical stochastic control theory and forward anticipated stochastic differential equations. Moreover, these problems will lead to a new kind of forward-backward stochastic differential equations (with anticipated equation as forward equation and delayed equation as backward equation). We will dedicate to study the existence and uniqueness of solutions for this kind of equations. And the partial differential equations relevant to stochastic anticipated differential equations or stochastic delayed differential equations will also be discussed by the results of the path-dependent PDE and non-linear Feynman-Kac formula.

本课题深入研究随机控制系统中的滞后、超前问题及其在实际问题中的应用。考虑正向带延迟系统的最优控制问题，借助随机分析、超前倒向随机微分方程等工具研究泛函形式的时滞系统的线性二次最优控制问题；带延迟的非零和随机微分对策问题等，并考虑理论结果在实际问题中的应用。在一些金融问题如带延迟的随机微分效用的启发下，创新性地提出倒向延迟控制系统，利用随机控制理论及新型的正向超前随机微分方程研究该类系统最优控制问题，包括最大值原理、线性二次最优控制等。同时，该类问题将衍生出一种新型的正倒向随机微分方程：正向为超前随机微分方程、倒向为延迟随机微分方程。我们将首先探讨该类方程解的存在唯一性问题，进而借助新近关于状态依赖的偏微分方程及非线性Feynman-Kac公式等理论，研究与倒向超前随机微分方程或倒向延迟随机微分方程相联系的偏微分方程理论。

本项目深入研究了随机控制系统中的滞后、超前问题及其在金融问题中的应用。以经典的最大值原理为核心，研究了状态变量和控制变量都有时滞的非零和随机微分对策问题，利用超前倒向随机微分方程作为对偶方程给出该类问题的最大值原理的必要性和充分性条件；研究了带松弛控制的时滞随机控制系统的最大值原理；利用随机发展方程来研究随机时滞系统的控制问题；研究了一类正向为超前随机微分方程，倒向为延迟随机微分方程的新型正倒向随机微分方程解的存在唯一性；研究了实际金融背景的金融投资问题、最优化等问题。得到了一批随机最优控制和对策领域国际前沿、国内领先的应用基础理论成果，并应用理论结果处理金融投资领域的实际问题。