The Gerber-Shiu Penalty Function for a Two-sided Renewal Risk Process Perturbed by Diffusion

We study the Gerber-Shiu discounted penalty function for a renewal risk model with random gains and perturbed by Brownian motion. Here the interarrival times have generalized Erlang distribution, and the process of random gains is a compound Poisson process with exponential jumps. We obtain the Laplace transform and a defective renewal equation for the discounted Gerber-Shiu penalty function, and when the claims have rational distributions, we give explicit expression for this function. An asymptotic result is derived for the probability of ruin when the distribution of claims is heavy-tailed. We provide some numerical results in the final section.