Non-exponential bounds for stop-loss premiums and ruin probabilities

We present an inventory of non-exponential bounds for ruin probabilities and stop-loss premiums in the general Sparre-Andersen model (renewal model) of risk theory. Various additional bounds are given if one assumes that the ladder height distribution F associated with the risk process belongs to a certain class of distributions, in particular if it is concave or it exhibits a (positive or negative) aging property. In most cases, these bounds are shown to improve existing ones in the literature and/or possess the correct asymptotic behaviour when the distribution F is subexponential. Since in the classical (compound Poisson) risk model the ladder height distribution is always concave, all the bounds given in the paper are also valid for this model. Finally, in many cases the results of the paper are also valid for any compound geometric distribution.