Actuarial applications of some Lundberg type bounds

Abstract

Different kinds of renewal equations repeatedly arise in connection with renewal risk models and variations. It is often appropriate to use bounds instead of the general solution to the renewal equation due to the inherent complexity. For this reason, as a first approach to construction of bounds we employ a general Lundberg type methodology. Second, we focus specifically on exponential bounds, which have the advantageous feature of being closely connected to the asymptotic behavior (for large values of the argument) of the general solution. Third, we incorporate cumulative distribution functions directly into the bounds themselves. In particular, the results obtained are applied to the probability of ultimate ruin when the zero-modified compound geometric model for aggregate claims is assumed. This particular model may naturally be viewed as a discrete analog of the classical compound Poisson model and is of considerable interest in its own right. Finally, in an aggregate claims context, bounds for the stop-loss premium are derived.