Optimal Utility Reinsurance And Dividend Policy In The Risk Model With Capital Injections

In this paper we study the optimal dividend problem with capitalinjections in the difusion model.During each time of dividend paymen-t,fixed transaction costs and proportional transaction costs are needed.Such dividends changed into impulse control problems.In order to re-duce its risk,the insurance company usually use reinsurance,so we con-sider the excess of loss reinsurance. At the same time,when the surplusis negative,capitals should be injected so that the company never ruin.Our goal is to maximize the total expected discounted utility of divi-dend payment minus the discounted injection system.We obtained thequasi variational inequalities that the value function should fulfill,andthe optimal value function and the optimal strategy are obtained.