A time of ruin constrained optimal dividend problem for spectrally one-sided Lévy processes

We introduce a longevity feature to the classical optimal dividend problem by adding a constraint on the time of ruin of the firm. We consider De Finetti's problem for one-sided Lévy risk models in both scenarios with and without fix transaction costs. To characterize the solution to the aforementioned models we introduce the dual problem and show that the complementary slackness conditions are satisfied and therefore there is no duality gap. As a consequence the optimal value function can be obtained as the pointwise infimum of auxiliary value functions. Finally, we illustrate our findings with a series of numerical examples.