Optimal time-consistent reinsurance and investment strategies for multiple dependent types of insurance business and a unified investment framework

This paper proposes a new insurance model, which consists of the main insurance business and n types of insurance sub-business. We consider the dependence between the main insurance business and the n types of insurance sub-business, which is measured by the correlation of the number of their claims. To reduce claim risk, the insurer can simultaneously purchase reinsurance for the main insurance business and n types of insurance sub-business, respectively. In addition, we consider a unified investment framework, which includes typical diversified and concentrated investment patterns in the literature. The objective of the insurer is to find the optimal time-consistent reinsurance and investment strategies so as to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. By using stochastic control theory and the Hamilton–Jacobi–Bellman equation technique, we obtain the explicit optimal time-consistent reinsurance and investment strategies. Finally, numerical examples are given to illustrate the effects of model parameters on the optimal strategies, and analysis of the results reveals some interesting phenomena.