Optimality of Impulse Control Problem in Refracted Lévy Model with Parisian Ruin and Transaction Costs

Here, we investigate an optimal dividend problem with transaction costs, in which the surplus process is modeled by a refracted Lévy process and the ruin time is considered with Parisian delay. The presence of the transaction costs implies that the impulse control problem needs to be considered as a control strategy in such a model. An impulse policy which involves reducing the reserves to some fixed level, whenever they are above another, is an important strategy for the impulse control problem. Therefore, we provide sufficient conditions under which the above described impulse policy is optimal. Furthermore, we provide new analytical formulae for the Parisian refracted q-scale functions in the case of the linear Brownian motion and the Crámer–Lundberg process with exponential claims. Using these formulae, we show that, for these models, there exists a unique policy, which is optimal for the impulse control problem. Numerical examples are also provided.