An optimization dichotomy for capital injections and absolutely continuous dividend strategies

We consider an optimal stochastic control problem in which a firm's cash/surplus process is controlled by dividend payments and capital injections. Stockholders aim to maximize their dividend stream minus the cost of injecting capital, if needed. We consider absolutely continuous dividend policies subject to a level-dependent upper bound on the dividend rate while we allow for general capital injections behavior. We prove that the optimal strategy can only be of two types: dividends are paid according to a \textit{mean-reverting} strategy with capital injections performed each time the cash process reaches zero; or, dividends are paid according to another \textit{mean-reverting} strategy and no injection of capital is ever made, until ruin is reached. We give a complete solution to this problem and characterize this dichotomy by comparing (the derivatives of) the value functions at zero of two sub-problems. The first sub-problem is concerned solely with the maximization of dividends, while the second sub-problem is the corresponding bail-out optimal dividend problem for which we provide also a complete solution.