Time-consistent mean-variance asset-liability management in a regime-switching jump-diffusion market

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.