Optimal Dividend And Capital Injection Strategies With Random Delay

In this paper,we consider the optimal dividend policy with delayed capital injec-tions,and assume that the capital injection delay follows the exponential distribution.The focus of the policy is to find the optimal dividend and capital injection strategy to maximize the utility of dividends and capital.Since surplus process of the insurance company involves a mixed Poisson process,we use a stochastic differential equation to characterize the surplus process by adopting diffusion approximation techniques and give the value function under the utility criterion.When the value function is smooth,the quasi variational inequality is obtained by using the dynamic programming method.In this paper,we consider the value function from three different regions(the dividend area,the continuous area,and the capital injection area),considering the dividend and the capital injection process with random delay.Through the boundary conditions,we get the expression of the value function in different regions and the corresponding optimal s-trategy,and give the verification theorem.Then,we discussed the case of capital injection delays in some special cases and gave some interesting economic insights.