On the probability of ruin in a continuous risk model with two types of delayed claims

This article studies a risk model involving one type of main claims and two types of by-claims, which is an extension of the general risk model with delayed claims. We suppose that every main claim may not induce any by-claims or may induce one by-claim belonging to one of the two types of by-claims with a certain probability. In addition, assume that the by-claim and its associated main claim may occur at the same time and that the occurrence of the by-claim may be delayed. An integro-differential equation system for survival probabilities is derived by using two auxiliary risk models. The expression of the survival probability is obtained by applying Laplace transforms and Rouche theorem. Furthermore, we provide a method for solving the survival probability when the two by-claim amounts satisfy different exponential distributions. As a special case, an explicit expression of survival probability is given when all the claim amounts obey the same exponential distribution. Finally, numerical results are provided to examine the proposed method.