We consider the optimal financing (capital injections) and dividend payment problem for a Brownian motion model in the case of restricted dividend rates. The company has no obligation to inject capitals and therefore, the bankruptcy risk is present. Capital injections, if any, will incur both fixed and proportional transaction costs and dividend payments incur proportional transaction costs. The aim is to find the optimal strategy to maximize the expected present value of dividend payments minus the total cost of capital injections up to the time of bankruptcy. The problem is formulated as a mixed impulse-regular control problem. We address the problem via studying three cases of two auxiliary functions. We derive important analytical properties of the auxiliary functions and use them to study the value function and then identify the optimal control strategy. We show that the optimal dividend control is of threshold type and the optimal financing strategy prescribes to either never inject capitals or inject capitals only when the surplus reaches 0 with a fixed lump sum amount.

We consider the optimal dividend control problem to find an optimal strategy under the constraint that dividend rates is restricted such that the expected total discounted dividends are maximized for an insurance company. The evolution of the reserve is modeled by a diffusion process with drift and volatility coefficients modulated by an observable Markov chain. We consider the regime-switching threshold strategy which pays out dividends at the maximal possible rate when the current reserve is above some critical level dependent on the regime of the Markov chain at the time, and pays nothing when the reserve is below that level. We give sufficient conditions under which such type of strategy is optimal for the regime-switching model.