The equivalence of the infinite time ruin problems for positive and negative risk models

For the classical Andersen's model for both positive and negative capital risk amounts, we consider the influence of considering the amount of the initial claim occurring at time 0. Using the duality principle (Janssen, 1969), we establish the equivalence theorem showing that the probability of ruin for a positive model with the initial claim is also the probability of ruin for the usual associated dual negative model without considering the initial claim and vice versa. The connection with queuing theory is shown and the results are illustrated by a numerical example.