This paper considers the problem of optimal investment-reinsurance for the insurer and reinsurer under the constant elasticity of variance (CEV) model. It is assumed that the net claims process is approximated by a diffusion process, both the insurer and reinsurer can invest in risk-free assets and risky assets. We use the variance premium principle to calculate the premiums of the insurer and reinsurer, and the reinsurance proportion is constrained by the net profit condition. Our objective is to maximize the joint exponential utility of the insurer and reinsurer's terminal wealth for a fixed time. By solving the HJB equation, we obtain the explicit expressions of the optimal investment-reinsurance strategy and value function. We find that the optimal reinsurance strategy can be divided into many cases and is related to the risk aversion coefficient of the insurer and reinsurer, but independent of the price of risky assets. Furthermore, we give the proof of the verification theorem. Finally, we demonstrate a numerical analysis to explain the results.

This paper investigates the time-consistent robust optimal reinsurance problem for the insurer and reinsurer under weighted objective criteria. The joint objective criterion is obtained by weighting the mean-variance objectives of both the insurer and reinsurer. Specifically, we assume that the net claim process is approximated by a diffusion model, and the insurer can purchase proportional reinsurance from the reinsurer. The insurer adopts the loss-dependent premium principle considering historical claims, while the reinsurance contract still uses the expected premium principle due to information asymmetry. Both the insurer and reinsurer can invest in risk-free assets and risky assets, where the risky asset price is described by the constant elasticity of variance model. Additionally, the ambiguity-averse insurer and ambiguity-averse reinsurer worry about the uncertainty of parameter estimation in the model, therefore, we obtain a robust optimization objective through the robust control method. By solving the corresponding extended Hamilton-Jacobi-Bellman equation, we derive the time-consistent robust equilibrium reinsurance and investment strategy and corresponding value function. Finally, we examined the impact of various parameters on the robust equilibrium strategy through numerical examples.