A dynamic reinsurance theory
The authors consider three variants of surplus (proportional stop-loss) reinsurance: (1) a simple contract, (2) a contract where reinsurance covers a part of the loss between the retention and a multiple, and (3) a combination of proportional stop-loss on the claim severity and proportional stop-loss on the claim number process. In each case they give the evident formulae for the conditional expectation and the conditional variance of the risk retained by the ceding company. Also, in each case, they adapt a result of \textit{N. Bouleau} and \textit{D. Lamberton} [Stochastic Processes Appl. 33, No. 1, 131-150 (1989; Zbl 0743.60069)] to specify the continuously adjusted reinsurface fraction the insurer should use to minimize the variance of the conditional expectation of retained loss. The Bouleau and Lamberton (B\& L) work prescribes conditions for the existence of an optimal hedging strategy between a stock and a contingent claim of a particular form on the stock; their result also gives the expression (in terms of a Carré du Champ operator) for the optimal strategy. The authors demonstrate that these conditions are met when the stock is taken to be the conditional expectation of the (non- proportional) stop-loss reinsurance premium: the optimal strategy for the fraction retained above the retention limit follows immediately from B\& L. A discussion of the numerical evaluation of the various formulae and a number of illustrative examples of each of the three variants completes the paper.
conditional expectation of retained loss
scientific article; zbMATH DE number 96295