Optimal reinsurance and investment problem for an insurer and a reinsurer with jump-diffusion risk process under the Heston model

This paper focuses on the optimal management problem for a general insurance company which holds shares of an insurance company and a reinsurance company. The general company aims to maximize the expected exponential utility of the weighted sum of the insurance company’s and the reinsurance company’s terminal wealth. In our model, the risk process is assumed to follow a jump-diffusion risk model in which the surplus process is a compound Poisson risk process perturbed by a diffusion. The insurance company transfers part of the risk due to insurance claims via the proportional reinsurance to the reinsurance company. In addition, both the insurance company and the reinsurance company are allowed to invest in a risk-free asset and a risky asset whose price process is described by the Heston model, respectively. By applying the method of stochastic control, we establish the Hamilton–Jacobi–Bellman equation and obtain the optimal reinsurance and investment strategies for the insurance company and the reinsurance company. Furthermore, numerical simulations are provided to illustrate the effects of model parameters on the optimal reinsurance and investment strategies.