Optimal reinsurance‐investment problem for maximizing the product of the insurer’s and the reinsurer’s utilities under a CEV model

This paper studies the optimal reinsurance and investment problem considering both an insurer’s utility and a reinsurer’s utility. The claim process is assumed to follow a Brownian motion with drift, and the insurer can purchase proportional reinsurance from the reinsurer. The insurer is allowed to invest in a risk‐free asset and a risky asset whose price satisfies the constant elasticity of variance (CEV) model. In addition, the reinsurer is allowed to invest in a risk‐free asset to reduce the risk. Taking both the insurer and the reinsurer into account, this paper aims to maximize the expected product of the insurer’s and the reinsurer’s exponential utilities of terminal wealth. By solving the corresponding Hamilton–Jacobi–Bellman (HJB) equation, we derive the optimal reinsurance and investment strategies explicitly. Furthermore, we discuss the properties of the optimal strategies analytically. Finally, numerical simulations are presented to illustrate the effects of model parameters on the optimal strategies.