OPTIMAL STRATEGY FOR A DISCRETE RISK PROCESS IN CASE OF ORDERED CLAIM DISTRIBUTIONS

The paper concerns the problem of choosing the optimal strategy by the insurer to minimalize the infinite time ruin probability. The authoress considers the risk process which is discrete with respect both to the time and the state space. She assumes that the insurer is able to change dynamically the portfolio content at every moment the risk process is observed. It means there is a possibility to change the parameters which influence the claim size distribution. A modification of the risk process allows to establishy that the optimal strategy exists and is obtained by comparing ruin probabilities for fixed claim size distributions. For stochastically ordered distributions it is possible to present the form of the optimal strategy. It consists in choosing always this distribution which is at least in the sense of this order. However, the same policy is not optimal for distributions ordered by the covex order. There are two counterexamples in the paper. The first one is analytical and concerns the easy case which one can call simple random walk, the second one is numerical but is more realistic. The optimal strategy for any discrete claim size distributions can be obtained numerically. Howeber, the results are not always consistent with intuitive expectations. t