发表机构
Anhui University; The Hong Kong Polytechnic University; Southern University of Science and Technology; Southeast University（安徽大学; 香港理工大学; 南方科技大学; 东南大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对含马尔可夫链跳跃的随机线性二次控制问题，证明耦合微分Riccati方程唯一可解，推导开环最优控制的闭环表示，并应用于均值-方差投资组合选择，获得有效前沿。
AI 中文摘要
本文研究了一类具有马尔可夫链跳跃的扩散系统的随机线性二次（SLQ）控制问题。与传统的将扩散过程与马尔可夫链耦合的马尔可夫链扩散系统不同，我们将马尔可夫链的跳跃纳入状态方程中。这种建模方法能够有效捕捉系统在状态切换期间潜在的收益或损失。值得注意的是，在状态方程中引入马尔可夫链跳跃会增加相应的耦合微分Riccati方程（CDREs）的复杂性，从而使控制问题的可解性更具挑战性。在代价泛函一致凸的假设下，我们证明了相应CDREs的唯一可解性。在此基础上，我们推导了唯一开环最优控制的闭环表示。最后，我们将理论结果应用于具有马尔可夫链跳跃的马尔可夫链金融市场中的均值-方差投资组合选择问题，并得到了其有效前沿。
英文摘要
This paper investigates a stochastic linear-quadratic (SLQ) control problem for a regime-switching jump-diffusion system. Unlike traditional regime-switching diffusion systems that couple a diffusion process with a Markov chain, we incorporate the jumps of the Markov chain into the state equation. This modeling methodology effectively captures potential gains or losses of the system during the regime transitions. It should be noted that the introduction of Markov chain jumps into the state equation leads to increased complexity in the corresponding coupled differential Riccati equations (CDREs), thereby rendering the solvability of the control problem more challenging. Under the assumption that the cost functional is uniformly convex, we establish the unique solvability of the corresponding CDREs. Building upon this foundation, we derive a closed-loop representation for the unique open-loop optimal control. Finally, we apply our theoretical results to the mean-variance portfolio selection problem in a regime-switching financial market with Markov chain jumps and obtain its efficient frontier.
