Time-inconsistent view on a dividend problem with penalty

We consider the dividend maximization problem including a ruin penalty in diffusion environment. The additional term is motivated by constraint on strategies. Intentionally, we use different discount rates for dividends and penalty, which causes time-inconsistency. This allows to study types of constraints. For approximation classical surplus process derive an explicit equilibrium strategy associated value function. Inspired duality arguments, can identify particular such that given initial imposed fulfilled. Furthermore, illustrate our findings with numerical example.