Parisian ruin with a threshold dividend strategy under the dual Lévy risk model

Abstract

We consider the threshold dividend strategy where a company’s surplus process is described by the dual Lévy risk model. Namely, the company chooses to pay dividends at a constant rate only when the surplus is above some nonnegative threshold. Classically, such a company is referred to be ruined immediately when the surplus level becomes negative. Recently, researchers investigate the Parisian ruin problem where the company is allowed to operate under negative surplus for a predetermined period known as the Parisian delay. With the help of the fluctuation identities of spectrally negative Lévy processes, we obtain an explicit expression of the expected discounted dividends until Parisian ruin in terms of the relevant scale functions and certain probabilities that need to be evaluated for each specific Lévy process. The optimal threshold level under such a threshold dividend strategy is deduced. Applications and numerical examples are given to illustrate the theoretical results and examine how the expected discounted aggregate dividends and the optimal threshold level change in response to different Parisian delays.