Time-consistent reinsurance-investment strategy for an insurer and a reinsurer with mean-variance criterion under the CEV model

This paper is devoted to derive the time-consistent reinsurance-investment strategy for an insurer and a reinsurer under mean-variance criterion. We aim to maximize the weighted sum of the insurer's and the reinsurer's objectives with different risk averse coefficients. The claim process is assumed to follow a Brownian motion with drift and the insurer can purchase proportional reinsurance from the reinsurer. Moreover, both the insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset, respectively. In order to reduce risks, the insurer and the reinsurer can invest in different risky assets whose price processes are described by the constant elasticity of variance (CEV) models. Consideration of the CEV model and the profits of both the insurer and the reinsurer, the proof of the verification theorem becomes difficult and the solution becomes complex. We first formulate a general problem and prove the verification theorem. By solving an extended Hamilton-Jacobi-Bellman (HJB) equation, we obtain the time-consistent reinsurance-investment strategy and the corresponding value function explicitly. Finally, sensitivity analysis and numerical simulation are presented to show the effects of parameters on the time-consistent strategy and illustrate the economic meaning.