On the time and aggregate claim amount until the surplus drops below zero or reaches a safety level in a jump diffusion risk model

We consider a two-barrier renewal risk model assuming that insurer's income is modeled via a Brownian motion, and the surplus is inspected only at claim arrival times. We are interested in the joint distribution of the time, number of claims and the total claim amount until the surplus process falls below zero (ruin) or reaches a safety level. We obtain a general formula for the respective joint generating function which is expressed via the distributions of the undershoot (deficit at ruin) and the overshoot (surplus exceeding safety level). We offer explicit results in the classical Poisson model, and we also study a more general renewal model assuming mixed Erlang distributed claim amounts and inter-arrival times. Our methodology is based on tilted measures and Wald's likelihood ratio identity. We finally illustrate the applicability of our theoretical results by presenting appropriate numerical examples in which we derive the distributions of interest and compare them with the ones estimated using Monte Carlo simulation.