Optimal insurance strategy in a risk process under a safety level imposed on the increments of the process

The problem of designing an optimal insurance strategy in a modification of the risk process with discrete time is investigated. This model introduces stage-by-stage probabilistic constraints (Value-at-Risk (VaR) constraints) on the insurer's capital increments during each stage. Also, the set of admissible insurances is determined by a safety level reflecting a ‘good’ or ‘bad’ capital increment at the previous stage. The mathematical expectation of the insurer's final capital is used as the objective functional. The total loss of the insurer at each stage is modeled by the Gaussian (normal) distribution with parameters depending on a seded loss function (or, in other words, an insurance policy) selected. In contrast to traditional dynamic optimization models for insurance strategies, the proposed approach allows to construct the value functions (and hence the optimal insurance policies) by simply solving a sequence of static insurance optimization problems. It is demonstrated that the optimal seded loss function at each stage depends on the prescribed value of the safety level: it is either a stop-loss insurance or conditional deductible insurance having a discontinuous point. In order to reduce ex post moral hazard, we also investigate the case, where both parties in an insurance contract are obligated to pay more for a larger realization of loss. This leads to that the optimal seeded loss functions are either stop-loss insurances or unconditional deductible insurances.