Time-Consistent Investment-Reinsurance Strategies Towards Joint Interests of the Insurer and the Reinsurer Under CEV Models

The present paper studies the time-consistent solutions to an investment-reinsurance problem under a mean-variance framework. The paper is distinguished from other literature by taking into account the interests of both the insurer and reinsurer jointly. The claim process of the insurer is governed by a Brownian motion with a drift. A proportional reinsurance treaty is considered and the premium is calculated according to the expected value principle. Both the insurer and the reinsurer are assumed to invest in a risky asset, which is distinct for each other and driven by a constant elasticity of variance model. The optimal decision is formulated on a weighted sum of the insurer’s and reinsurer’s surplus processes, where the weight can be viewed as a regularization parameter to balance the importance of each party in the decision. Upon a verification theorem, which is established with a formal proof for a more general problem, explicit solutions are obtained for the proposed investment-reinsurance model. Moreover, numerous mathematical analysis and numerical examples are given to demonstrate those derived results as well as the economic meanings behind.