A connection between the discounted and non-discounted expected penalty functions in the Sparre Andersen risk model

In this paper, we study the Gerber-Shiu expected discounted penalty function in a Sparre Andersen risk model probabilistically. This is implemented through the modified random walk defined herein. As a result, we derive explicit formulas for the discounted probability of ruin and the discounted distribution of the deficit at ruin when the distribution of the interclaim times is arbitrary and the distribution of claim sizes is phase-type. In addition, we provide iterative schemes for evaluating the formulas.