We study the dual model with capital injection under the additional condition that the dividend strategy is absolutely continuous. We consider a refraction–reflection strategy that pays dividends at the maximal rate whenever the surplus is above a certain threshold, while capital is injected so that it stays non-negative. The resulting controlled surplus process becomes the spectrally positive version of the refracted–reflected process recently studied by Pérez and Yamazaki (2015). We study various fluctuation identities of this process and prove the optimality of the refraction–reflection strategy. Numerical results on the optimal dividend problem are also given.
