On the spectrally negative Levy risk process with mixed dividends and capital injections

In this article, we study the spectrally negative Levy risk process with mixed dividends and capital injections. Given a spectrally negative Levy risk process with two boundaries and a sequence of random observation times, it is reflected at the lower boundary 0 and refracted at the upper boundary b(>0). Additionally, whenever the observed surplus is above b, all the excess is paid out as dividends. Using the scale function, we compute the expected discounted dividends and the expected discounted capital injections. The optimal dividend strategy is also discussed on maximizing the expected dividends subtracted by the discounted cost of capital injections. Numerical examples are presented to illustrate our results.