A Lévy risk model with ratcheting and barrier dividend strategies

<p style='text-indent:20px;'>The expected present value of dividends is one of the classical stability criteria in actuarial risk theory. In this paper, we consider the two-layer <inline-formula><tex-math id="M1">\begin{document}$ (a, b) $\end{document}</tex-math></inline-formula> dividend strategy when the risk process is modeled by a spectrally negative Lévy process, such a strategy has an increasing dividend rate when the surplus exceeds level <inline-formula><tex-math id="M2">\begin{document}$ a>0 $\end{document}</tex-math></inline-formula>, and all of the excess over <inline-formula><tex-math id="M3">\begin{document}$ b>a $\end{document}</tex-math></inline-formula> as lump sum dividend payments. Using fluctuation identities and scale functions, we obtain explicit formulas for the expected net present value of dividends until ruin and the Laplace transform of the time to ruin. Finally, numerical illustrations are present to show the impacts of parameters on the expected net present value.</p>