We study de Finetti's optimal dividend problem, also known as the optimal harvesting problem, in the dual model. In this model, the firm value is affected both by continuous fluctuations and by upward directed jumps. We use a fixed point method to show that the solution of the optimal dividend problem with jumps can be obtained as the limit of a sequence of stochastic control problems for a diffusion. In each problem, the optimal dividend strategy is of barrier type, and the rate of convergence of the barrier and the corresponding value function is exponential.
"The optimal dividend problem in the dual model." Adv. in Appl. Probab. 46 (3) 746 - 765, September 2014. https://doi.org/10.1239/aap/1409319558