Martingales and the distribution of the time to ruin

We determine the ultimate ruin probability and the Laplace transform of the distribution of the time to ruin in the classical risk model, where claims arrive according to a renewal process, with waiting times that are of phase-type, while the claims themselves follow a distribution with a Laplace transform that is a rational function. The main tools are martingales, the optional sampling theorem and results from the theory of piecewise deterministic Markov processes.