On the Expected Discounted Penalty Function for a Risk Model with Thinning Process

Abstract

This paper studies the expected discounted penalty function for a risk model in which the arrival of insurance policies is a Poisson process and the process of claim occurring is -thinning process. Using backward differential argument, we derive the integro-differential equation satisfied by the expected discounted penalty function when the stochastic discount interest process is perturbed by standard Wiener process and Poisson-Geometric process. 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.