On the discounted distribution functions of the surplus process perturbed by diffusion

In this paper, we derive explicit expressions for the discounted joint and marginal distribution functions of the surplus immediately prior to the time of ruin and the deficit at the time of ruin, and for the discounted distribution function of the amount of the claim causing ruin, based on the surplus process of ruin theory with an independent diffusion process. Furthermore, we show that these distribution functions satisfy defective renewal equations.