On the moments of dividends and capital injections under a variant type of Parisian ruin

This paper considers a risk model driven by a spectrally negative Lévy process, where any surplus above $b$ ($0<b<\infty$) is deducted away as dividends and any deficit is covered by injected capitals/raised money. For such a risk model, we define a variant of Parisian ruin time as the first time that the surplus process stays continuously below $a$ ($0<a<b<\infty$) for a time interval with length larger than some pre-specified exponential random variable that is marked on this time interval. A recursive formula for the moments of the Net Present Value (NPV) of dividends paid until Parisian ruin is provided. The expected NPV of capitals injected until the Parisian ruin time is also characterized compactly in terms of the scale functions of the underlying process.