Optimal Periodic Dividend and Capital Injection Strategies for Diffusion Models with Restricted Dividend Rates and Transaction Costs

We consider the optimal joint dividend and capital injection strategy with proportional and fixed costs. The main feature of our diffusion model is that dividend can only start at the arrival times of an exogenous uncontrolled Poisson process (signal), and the dividend rate is restricted with a bounded interval. The objective is to find the control which maximizes the expected total discounted dividends less the expected discounted costs of capital injections. We also give some numerical illustrations to optimal strategies, which indicate that the optimal threshold levels are increasing functions of exogenous parameters.