On capital injections and dividends with tax in a diffusion approximation

We consider a diffusion approximation to a risk process with dividends and capital injections. Tax has to be paid on dividends, but capital injections lead to an exemption from tax. That is, tax is only paid for the aggregate excess of dividends over the capital injections. The value of a strategy is the expected value of the discounted dividend payments after tax minus the discounted capital injections. We solve the problem and show that the optimal dividend strategy is a barrier strategy.