Optimal insurance strategies in a risk process with restrictions on policyholder risks

We consider the optimal choice problem by a risk-bearing function for an insurer to divide risks between him and his clients in a dynamic insurance model, the so-called Cramer-Lundberg risk process. In this setting, we take into account restrictions imposed on policyholder risks, either on the mean value or a constraint with probability one. We solve the optimal control problem on an infinite time interval for the optimality criterion of the stationary coefficient of variation. We show that in the model with a restriction on average risk the stop-loss insurance strategy will be most profitable. For a probability one restriction, the optimal insurance is a combination of a stop-loss strategy and a deductible. We show that these results extend to a number of problems with other optimality criteria, e.g., the problems of maximizing unit utility and minimizing the probability of deviating from the mean value.