论文标题
随机Volterra积分方程的线性季度最佳控制：因果状态反馈和路径依赖性Riccati方程
Linear-Quadratic Optimal Controls for Stochastic Volterra Integral Equations: Causal State Feedback and Path-Dependent Riccati Equations
论文作者
论文摘要
考虑到正向随机Volterra积分方程（简称FSVIE）的线性季度最佳控制问题。在通常的凸度条件下，存在开环的最佳控制，可以以最佳系统，FSVIE的耦合系统和II型向后SVIE（简称BSVIE）来表征。为了获得开放环最佳控制的因果状态反馈表示，引入了与操作员值函数的路径依赖性riccati方程，通过该方程，可以通过该方程将最佳系统解耦。在解耦过程中，引入了III型BSVIE，其适应性解决方案可用于表示相应的II型BSVIE的改编的M-solution。在某些条件下，证明路径依赖性的Riccati方程允许一个独特的解决方案，这意味着找到了最佳系统的解耦场。因此，构建了开环最佳控制的因果状态反馈表示。另一个有趣的发现是，当控件仅出现在扩散项中而不在状态系统的漂移项中时，因果状态反馈就会减少到马尔可夫状态反馈。
A linear-quadratic optimal control problem for a forward stochastic Volterra integral equation (FSVIE, for short) is considered. Under the usual convexity conditions, open-loop optimal control exists, which can be characterized by the optimality system, a coupled system of an FSVIE and a Type-II backward SVIE (BSVIE, for short). To obtain a causal state feedback representation for the open-loop optimal control, a path-dependent Riccati equation for an operator-valued function is introduced, via which the optimality system can be decoupled. In the process of decoupling, a Type-III BSVIE is introduced whose adapted solution can be used to represent the adapted M-solution of the corresponding Type-II BSVIE. Under certain conditions, it is proved that the path-dependent Riccati equation admits a unique solution, which means that the decoupling field for the optimality system is found. Therefore a causal state feedback representation of the open-loop optimal control is constructed. An additional interesting finding is that when the control only appears in the diffusion term, not in the drift term of the state system, the causal state feedback reduces to a Markovian state feedback.
