Abstract: This paper investigates an optimal time-consistent reinsurance and investment problem with $n$ dependent insurance businesses for an insurance company. The insurance company is allowed to purchase reinsurance for reducing claim risk and invest in the financial market for increasing wealth. The financial market consists of one risk-free asset and $n$ correlated risky assets, whose price processes are described by diffusion processes. Then, we establish the wealth process of the insurance company by using the stochastic analysis theory. Our main goal is to find an optimal time-consistent reinsurance and investment strategy so as to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. By applying the stochastic control and stochastic dynamic programming techniques, we establish the extended Hamilton-Jacob-Bellman (HJB) equation. Explicit solutions for the optimal reinsurance and investment strategy as well as the corresponding value function are obtained by solving the extended HJB equation. Finally, numerical experiments illustrate the effects of model parameters on the optimal time-consistent reinsurance and investment strategy.

Key words: time consistency, investment, reinsurance, stochastic control, HJB equation