An optimal dividend policy with delayed capital injections

This work focuses on finding optimal dividend payment and capital injection policies to maximize the present value of the difference between the cumulative dividend payment and the possible capital injections with delays. Starting from the classical Cramér-Lundherg process, using the dynamic programming approach, the value function obeys a quasi-variational inequality. With delays in capital injections, the company will be exposed to the risk of financial ruin during the delay period. In addition, the optimal dividend payment and capital injection strategy should balance the expected cost of the possible capital injections and the time value of the delay period. In this paper, the closed-form solution of the value function and the corresponding optimal policies are obtained. Some limiting cases are also discussed. A numerical example is presented to illustrate properties of the solution. Some economic insights are also given.