This paper investigates the time-consistent optimal control of a mean–variance asset-liability management problem in a regime-switching jump-diffusion market. The investor (a company) is investing in the market with one risk-less bond and one risky stock while subject to an uncontrollable liability. The risky stock and the liability processes are discontinuous with correlated jumps and modulated by a continuous-time observable Markov chain that represents the market state. This hybrid model can capture both long-term and short-term effects on the investing process resulting from the market movements and unexpected events. Under a game-theoretic framework, we derive the regime-switching jump-diffusion version of the extended Hamilton–Jacobi–Bellman (HJB) system as well as the verification theorem, based on which we obtain the closed-form equilibrium control and equilibrium value function in terms of five systems of ordinary differential equations by solving the extended HJB equation. Finally, a numerical analysis investigates the influence of changes in the model parameters on our solution, and it is discovered that regime switching and jump diffusion both have great effect on the investment and therefore should be considered in conjunction.

中文翻译：

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政权转换跳跃扩散市场中的时间一致均值方差资产负债管理

本文研究了政权转换跳跃扩散市场中均值方差资产负债管理问题的时间一致最优控制。投资者（一家公司）以一种无风险债券和一种风险股票投资于市场，同时承担无法控制的负债。风险股票和负债过程是不连续的，具有相关的跳跃，并由代表市场状态的连续时间可观察马尔可夫链调制。这种混合模型可以捕捉市场变动和意外事件对投资过程的长期和短期影响。在博弈论框架下，我们推导出了扩展的 Hamilton-Jacobi-Bellman (HJB) 系统的状态转换跳跃扩散版本以及验证定理，在此基础上，我们通过求解扩展的HJB方程，得到了五个常微分方程组的闭式平衡控制和平衡值函数。最后，数值分析研究了模型参数的变化对我们的解决方案的影响，发现状态转换和跳跃扩散都对投资有很大影响，因此应结合考虑。