Optimal investment problem for an insurer and a reinsurer

This paper focuses on the optimal investment problem for an insurer and a reinsurer. The insurer’s and reinsurer’s surplus processes are both approximated by a Brownian motion with drift and the insurer can purchase proportional reinsurance from the reinsurer. In addition, both the insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset. We first study the optimization problem of minimizing the ruin probability for the insurer. Then according to the optimal reinsurance proportion chosen by the insurer, we study two optimal investment problems for the reinsurer: the problem of maximizing the exponential utility and the problem of minimizing the ruin probability. By solving the corresponding Hamilton-Jacobi-Bellman (HJB) equations, we derive optimal strategies for both the insurer and the reinsurer explicitly. Furthermore, we find that the reinsurer’s optimal strategies under the two cases are equivalent for some special parameters. Finally, numerical simulations are presented to illustrate the effects of model parameters on the optimal strategies. Key–Words: Proportional reinsurance, Optimal investment, For a reinsurer, Hamilton-Jacobi-Bellman (HJB) equation, Exponential utility maximization, Ruin probability minimization