Choosing the Instant of Insurance Commencement

Abstract

Consideration was given to a continuous insurance model where the client is insured at a random time instant depending on the degree of wear of the insured object. The insurer designates the insurance premium and the insurance recovery that is paid to the client provided that the object was insured prior to the insured accident (failure). It was proved that the client bears the minimum losses if it concludes the insurance contract when the degree of wear (proportional to the danger of failure) reaches a level equal numerically to the ratio of the insurance premium to the insurance recovery. Expecting this optimal behavior of the client, the insurer seeks the most advantageous ratio. If there is no penalty for nonconcluded contract (the insured accident occurs before conclusion of the contract), then the greater the ratio, the greater on the average the relative insurer's profit from the enterprise. If there exists a penalty, then there exists a ratio of the insurance premium to the insurance recovery that is most advantageous to the insurer. The problem of determining the optimal ratio is solvable numerically in the class of monotone continuous semi-Markovian processes for which determination of the optimal level comes to solving a transcendental equation with functions defined by the process parameters.