马尔可夫切换模型的不定随机线性二次最优控制问题
Indefinite Stochastic Linear-Quadratic Optimal Control Problem for a Markov Regime-Switching Model
浏览论文内容
中文总结 AI 辅助
本文针对参数受马尔可夫切换影响的不定随机线性二次控制问题，引入松弛补偿器扩展其适用范围，推导最优控制的开环与闭环形式，并通过资产配置问题验证了理论框架的有效性与实用价值。
中文摘要 AI 辅助
本文研究参数受马尔可夫切换影响的不定随机线性二次（SLQ）控制问题。基于SLQ问题的适定性，引入松弛补偿器，将SLQ控制问题从正定情形扩展至不定情形。针对不定框架下的无约束与有约束控制情形，分析相应的随机哈密顿系统，推导最优开环控制。进一步研究无约束与有约束控制情形下的相关黎卡提方程，推导最优控制的闭环反馈形式。通过无约束与非负控制约束下的股票-债券资产配置问题说明理论结果，数值模拟验证了该理论框架的有效性，证明其在求解含马尔可夫切换的复杂随机控制问题中的实用价值。
英文摘要
This paper investigates an indefinite stochastic linear-quadratic (SLQ) control problem with parameters subject to Markov regime-switching. Based on the well-posedness of the SLQ problem, we introduce a relaxed compensator that extends SLQ control problems from the positive definite case to the indefinite case. We analyze the corresponding stochastic Hamiltonian system for both unconstrained and constrained control cases under the indefinite framework and derive the corresponding optimal open-loop controls. We further investigate the associated Riccati equations for both unconstrained and constrained control cases and derive the closed-loop feedback forms of optimal controls. We illustrate the theoretical results with an equity-bond asset allocation problem under un-constrained and non-negative control constraints. Numerical simulations validate the effectiveness of the theoretical framework and demonstrate its practical value in solving complex stochastic control problems with Markov regime-switching.
