Optimal Dividend and Capital Injection Problem with Random Delay

This article considers the optimal dividend policy with delayed capital injections, and assumes that the capital injection delay follows the exponential distribution. We aim to find the optimal dividend and capital injection strategies to maximize the utility of dividend 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 then we obtain the value function under the utility criterion. When the value function is smooth, the quasi variational inequality is obtained by using the dynamic programming principle. In this paper, we consider the value function from three different regions (the dividend area, the continuous area and the capital injection area). Through the boundary conditions, we derive the expression of the value function in different regions and present the verification theorem. A numerical example is presented to illustrate the effects of the capital injection delay under different parameters.