Optimality of impulse control problem in refracted L\'evy model with Parisian ruin and transaction costs

In this paper we investigate an optimal dividend problem with transaction costs, where the surplus process is modelled by a refracted L\'evy process and the ruin time is considered with Parisian delay. Presence of the transaction costs implies that one need to consider the impulse control problem as a control strategy in such model. An impulse policy $(c_1,c_2)$, which is to reduce the reserves to some fixed level $c_1$ whenever they are above another level $c_2$ is an important strategy for the impulse control problem. Therefore, we give sufficient conditions under which the above described impulse policy is optimal. Further, we give the new analytical formulas for the Parisian refracted $q$-scale functions in the case of the linear Brownian motion and the Cr\'amer-Lundberg process with exponential claims. Using these formulas we show that for these models there exists a unique $(c_1, c_2)$ policy which is optimal for the impulse control problem. Numerical examples are also provided.