INDEFINITE STOCHASTIC LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS WITH RANDOM COEFFICIENTS: CLOSED-LOOP REPRESENTATION OF OPEN-LOOP OPTIMAL CONTROLS

• 成果类型: Article
• 署名作者: Sun, Jingrui; Xiong, Jie; Yong, Jiongmin
• 署名单位: Southern University of Science & Technology; State University System of Florida; University of Central Florida
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1595
• 发表日期: 2021
• 页码: 460-499
• 关键词: variance portfolio selection; differential-games; regulators

摘要

This paper is concerned with a stochastic linear-quadratic optimal control problem in a finite time horizon, where the coefficients of the control system are allowed to be random, and the weighting matrices in the cost functional are allowed to be random and indefinite. It is shown, with a Hilbert space approach, that for the existence of an open-loop optimal control, the convexity of the cost functional (with respect to the control) is necessary; and the uniform convexity, which is slightly stronger, turns out to be sufficient, which also leads to the unique solvability of the associated stochastic Riccati equation. Further, it is shown that the open-loop optimal control admits a closed-loop representation. In addition, some sufficient conditions are obtained for the uniform convexity of the cost functional, which are strictly more general than the classical conditions that the weighting matrix-valued processes are positive (semi-) definite.