Ruin probability in the delayed renewal risk model perturbed by a diffusion process

In this paper, we consider the risk model perturbed by an independent diffusion process with a time delay in the arrival of the first claim. We derive the distribution of the counting process of the risk process, the integro-differential equations of the ruin probabilities and generalize its defective renewal equations. With claim amounts following exponential and mixed exponential distributions, explicit expressions and asymptotic properties of the ruin probabilities are derived. Numerical illustrations of the ruin probabilities are proposed when claim amounts are exponentially and mixed exponentially distributed. We extend the results to the case of the delayed renewal risk model with exchangeable risks which captures the possible dependence between inter-arrival times of the claims and derive the associated ruin probabilities.