Optimal dividend payment strategies with debt constraint in a hybrid regime-switching jump–diffusion model

This paper develops an optimal dividend policy for an insurance company, whose asset and liability have different dynamics, and whose surplus follows a regime-switching jump–diffusion process. The insurer aims to maximize the expected total discounted value of dividends paid out under the debt management constraint. By using the dynamic programming principle, a generalized system of integro-differential Hamilton–Jacobi–Bellman equations is derived. Moreover, this paper studies two cases where the liability of the insurer follows a uniform claim density and a generalized Pareto density in a two-regime economy, respectively. Closed-form solutions of the value functions and the optimal dividend payment strategies are obtained in both cases. A numerical example is provided to illustrate the relationship between the key parameters in the model and the value functions, as well as some interesting economic insights.