Upper and lower bounds on infinite time ruin probabilities in case of constraints on claim size distributions

Abstract

Probabilities of ruin (or non-ruin) are solutions of differential or integro-differential equations. Solving these kinds of equations analytically and/or numerically causes a lot of mathematical difficulties. In addition there exists a practical problem of determining, estimating or guessing the distribution of the risk. A realistic way to deal with this problem consists in deriving upper and lower bounds for the ruin probability in case of incomplete information on the distribution F. The present contribution is inspired by and generalises a result of G. Taylor who uses the concept of ordering of risks to order ruin probabilities. We show how some of the results obtained by F. De Vylder for deriving sharp bounds on the stop-loss premium in case of incomplete information can be applied to the evaluation of practical bounds on infinite time ruin probabilities.