Insurance risk with variable number of policies

In this article we consider an insurance company selling life insurance policies. New policies are sold at random points in time, and each policy stays active for an exponential amount of time with rate mu, during which the policyholder pays premiums continuously at rate r. When the policy expires, the insurance company pays a claim of random size. The aim is to compute the probability of eventual ruin starting with a given number of policies and a given level of insurance fund. We establish the remarkable result that the ruin probability is identical to the one in the standard compound Poisson model where the insurance fund increases at constant rate r and claims occur according to a Poisson process with rate mu.