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In the spectrally negative Lévy risk model, we investigate the absolutely continuous dividend problem with a general discount function, which results in a time-inconsistent control problem. Under the assumptions of a time value of ruin and an exponential time horizon, we study the equilibrium dividend strategies within a game theoretic framework for the return function composed by the discount expected dividend before the ruin. Using the technique of extended Hamilton-Jacobi-Bellman system of equations, we show the verification theorem and give the property of return function. For a mixture of exponential discount function, we obtain closed-form equilibrium dividend strategies and the corresponding equilibrium value functions in both a Cramér–Lundberg model and its diffusion approximation. In addition, some numerical examples are presented to discuss the impacts of some parameters on the control problem.

Equilibrium dividend strategies for spectrally negative Lévy processes with time value of ruin and random time horizon