Excess of loss reinsurance and Gerber’s inequality in the Sparre Anderson model

Assuming that the reinsurance premium is calculated according to the expected value principle we study an upper bound for the probability of ruin in finite horizon, as function of the excess of loss retention limit. The upper bound used is an extension proved by Grandell [Aspects of Risk Theory, Springer, New York, 1991] of Gerber’s bound, see Gerber [Martingales in risk theory, Mitteilungen der Vereinigung Schweizerischer Versicherungsmathematiker, 1973, pp. 205–216], for the Sparre Anderson model [On the collective theory of risk in the case of contagious between the claims, in: Proceedings of the Transactions on XV International Congress of Actuaries, New York, 1957].