  We consider an optimal control problem of a property insurance company with
proportional reinsurance strategy. The insurance business brings in catastrophe
risk, such as earthquake and flood. The catastrophe risk could be partly
reduced by reinsurance. The management of the company controls the reinsurance
rate and dividend payments process to maximize the expected present value of
the dividends before bankruptcy. This is the first time to consider the
catastrophe risk in property insurance model, which is more realistic. We
establish the solution of the problem by the mixed singular-regular control of
jump diffusions. We first derive the optimal retention ratio, the optimal
dividend payments level, the optimal return function and the optimal control
strategy of the property insurance company, then the impacts of the catastrophe
risk and key model parameters on the optimal return function and the optimal
control strategy of the company are discussed.
