Abstract
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.
Highlights
For many years, applied mathematicians have been trying to create models that allow describing reality in terms of mathematics
The last part of this paper is an example section, where we will give new analytical formulae for the Parisian refracted scale functions in the case of the linear Brownian motion and the Crámer–Lundberg process with exponential claims. Using these formulae, we will show that for these models, there exists a unique impulse policy which is optimal for the impulse control problem
It is assumed that the distance between c1 and c2 must be greater than β because after paying the transaction costs, there must be something left for the shareholders
Summary
For many years, applied mathematicians have been trying to create models that allow describing reality in terms of mathematics. Many alternatives have appeared in the literature, for example, the so-called Parisian ruin model, which is considered in this study In this approach, we say that the company announces bankruptcy if the risk process goes below zero (or the so-called red zone) and stays there longer than a certain fixed time r > 0. The last part of this paper is an example section, where we will give new analytical formulae for the Parisian refracted scale functions in the case of the linear Brownian motion and the Crámer–Lundberg process with exponential claims Using these formulae, we will show that for these models, there exists a unique impulse policy which is optimal for the impulse control problem.
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