Bounds for the probability and severity of ruin in the Sparre Andersen model

We propose a method for improving existing upper and lower bounds for ruin probabilities in the Sparre Andersen model. The method is based on the renewal equation for the probability of ruin in that model and may be used recursively. We discuss how the method can be used more generally for solutions of renewal equations, provided that the function in question is monotone and initial (analytic) bounds for it are available. We illustrate this by considering the severity of ruin in the Sparre Andersen model, for which we obtain initial lower and upper bounds; the application of the method is then straightforward.