Laguerre Series Expansion for Scale Functions and Its Applications in Risk Theory

We propose an exact explicit closed-form Laguerre series expansion formula to compute the q-scale function of a spectrally negative Lévy process (SNLP), and other functions associated to the scale function for the first time. The proposed closed-form formula for the scale function has many applications in applied probability and in particular in the Lévy insurance risk theory. We shall show that the new series expansion formulas can be used to express the expected discounted penalty functions, the moments of the present value of total dividend payments as well as the time value of Parisian ruin in the Lévy risk models.