Sharp conditions for certain ruin in a risk process with stochastic return on investments

We consider a classical risk process compounded by another independent process. Both of these component processes are assumed to be Lévy processes. Sharp conditions are given on the parameters of these two components to ensure when ruin is certain, and also when the time of ruin is of exponential type. It is shown that under some weak conditions, these problems depend only on the compounding process. When ruin is not certain, it is shown in Paulsen (1993) that the ruin probability depends on the distribution function of a certain present value, and an integro-differential equation for the characteristic function is found there in the special case when the two component Lévy processes have only a finite number of jumps on any finite time interval. We generalize this equation to the present case.