The compound Poisson risk model with a threshold dividend strategy

In this paper, we present the classical compound Poisson risk model with a threshold dividend strategy. Under such as strategy, no dividends are paid if the insurer's surplus is below certain threshold level. When the surplus is above this threshold level, dividends are paid at a constant rate that does not exceed the premium rate. Two integro-differential equations for the Gerber-Shiu discounted penalty function are derived and solved. The analytic results obtained are utilized to derive the probability of ultimate ruin, the time of ruin, the distribution of the first surplus drop below the initial level, and the joint distributions and moments of the surplus immediately before ruin and the deficit at ruin. The special cases where the claim size distribution is exponential and a combination of exponentials are considered in some detail.