Differentiability and Asymptotic Properties of Gerber-Shiu Function Associated with Absolute Ruin Time for a Risk Model with Random Premium Income

In this paper, the differentiability and asymptotic properties of Gerber-Shiu expected discounted penalty function (Gerber-Shiu function for short) associated with the absolute ruin time are investigated, where the risk model is given by classical risk model with additional random premium incomes. The additional random premium income process is specified by a compound Poisson process. A couple of integro-differential equations satisfied by Gerber-Shiu function are derived, several sufficient conditions which guarantee the second-order or third-order differentiability of Gerber-Shiu function are provided. Based on the differentiability results, when the individual claim and premium income are both exponential distribution, the previous integro-differential equations can be deduced into a third-order constant ordinary differential equation (ODE for short). With the standard techniques on ODE, we find the asymptotic behavior of absolute ruin probability when the initial surplus tends to infinity.