AI 中文总结
研究马尔可夫状态切换跳跃扩散系统无限时域随机线性二次控制问题，通过耦合代数黎卡提方程系统的稳定解给出最优控制反馈表示，并应用于终身财富跟踪问题得出最优投资策略。
AI 中文摘要
本文研究一类马尔可夫状态切换跳跃扩散系统在无限时域上的随机线性二次（SLQ）控制问题。与经典由马尔可夫链调制的扩散模型不同，状态过程会经历与马尔可夫链状态切换同步的突然跳跃。与传统泊松跳跃扩散模型不同，状态过程中的跳跃完全由马尔可夫链的状态转移引起。在此设定下，深入讨论了SLQ控制问题的闭环可解性，并通过耦合代数黎卡提方程（CAREs）系统的稳定解给出了最优控制的反馈表示。最后将结果应用于终身财富跟踪问题并得出相应最优投资策略。
英文摘要
This paper investigates a class of stochastic linear-quadratic (SLQ) control problems over an infinite horizon for Markov regime-switching jump-diffusion systems. Unlike classical diffusion models modulated by a Markov chain, we assume that the state process undergoes abrupt jumps that are synchronous with the regime switches of the Markov chain. In contrast to conventional Poisson jump-diffusion models, the jumps in the state process are entirely induced by the state transitions of the Markov chain, which can be interpreted as losses or gains of state process incurred during regime changes. Under this formulation, we thoroughly discuss the closed-loop solvability of the SLQ control problem and provide a feedback representation of the optimal control via the stabilizing solution of a system of coupled algebraic Riccati equations (CAREs). Finally, we further apply our results to a lifetime wealth tracking problem and derive the corresponding optimal investment strategy.
