On the Expected Discounted Penalty Function Associated with the Time of Ruin for a Risk Model with Random Income

This paper studies the expected discounted penalty function associated with the time of ruin for a risk model with stochastic premium. The premium process is no longer a linear function of time in contrast with the classical Cram\'er-Lundberg model. The aggregate premiums constitute a compound Poisson process which is also independent of the claim process. Integral equation for the penalty function is derived, which provides a unified treatment to the ruin quantities. Applications of the integral equation are given to the Laplace transform of the time of ruin, the deficit at ruin, the surplus immediately before ruin occurs. In some special cases with exponential distributions, closed form expressions for these quantities are obtained, which generalize some known results about the problems of ruin in Boikov (2003).