Linear-Quadratic Optimal Controls for Stochastic Volterra Integral Equations: Causal State Feedback and Path-Dependent Riccati Equations
Linear-Quadratic Optimal Controls for Stochastic Volterra Integral Equations: Causal State Feedback and Path-Dependent Riccati Equations
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DOI：
10.1137/22m1492696
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发表时间：
2022-04
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通讯作者：
Hanxiao Wang;J. Yong;Chao Zhou
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文献类型：
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作者：
Hanxiao Wang;J. Yong;Chao Zhou
研究一类正向随机沃尔泰拉积分方程（FSVIE）的线性二次型最优控制问题.在通常的凸性条件下，开环最优控制是存在的，它可以用最优性系统来描述，即一个FSVIE和一个Ⅱ型后向SVIE（BSVIE，简称）的耦合系统。为了得到开环最优控制的因果状态反馈表示，引入了一个算子值函数的路径依赖Riccati方程，通过该方程可以解耦最优系统。在解耦过程中，引入了一个III型BSVIE，其自适应解可以用来表示相应II型BSVIE的自适应M解。在一定条件下，证明了路径依赖的Riccati方程存在唯一解，即找到了最优系统的解耦域，从而构造了开环最优控制的因果状态反馈表示.另一个有趣的发现是，当控制只出现在扩散项，而不是在状态系统的漂移项，因果状态反馈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.