Randomized dividends in the compound binomial model with a general premium rate

In this paper, we consider the compound binomial model with a multi-threshold dividend structure and randomized dividend payments. Using the roots of a generalization of Lundberg's fundamental equation and the general theory on difference equations, we derive an explicit expression for the Gerber-Shiu discounted penalty function with any initial surplus u (u is an element of N). This result generalizes the main result of Tan & Yang (2006) regarding the recursive calculation of some Gerber-Shiu functions in a special class of risk models, namely the compound binomial model with a unit premium and a single threshold dividend structure. Finally, an explicit expression is also derived for the expected discounted dividend payments before ruin.