On the renewal risk process with stochastic interest

In this paper, we consider the renewal risk process with stochastic interest. For this risk process, we derive exact expressions and integral equations for the Gerber–Shiu expected discounted penalty function and the ultimate ruin probability. When the interest is received at a constant rate and the inter-occurrence times of claims follow an Erlang distribution, we obtain an integro-differential equation for the expected discounted penalty function. We also give lower and upper bounds for the ultimate ruin probability. Finally, we present exact expressions for the discounted density associated with the expected discounted penalty function in two special cases of stochastic interest processes.