Before revision: We consider the classical optimal dividend payments problem under the Cram\'er-Lundberg model with exponential claim sizes subject to a constraint on the time of ruin (P1). We use the Lagrangian dual function which leads to an auxiliary problem (P2). For this problem, given a multiplier \Lambda, we prove the uniqueness of the optimal barrier strategy and we also obtain its value function. Finally, we prove that the optimal value function of (P1) is obtained as the point-wise infimum over \Lambda of all value functions of problems (P2) .
After revision: We consider the classical optimal dividend payments problem under the Cram\'er-Lundberg model with exponential claim sizes subject to a constraint on the time of ruin . We introduce the dual problem and show that the complementary slackness conditions are satisfied, thus there is no duality gap. Therefore the optimal value function of (P1) is obtained as the point-wise infimum over \Lambda of all value functions of problems (P2) .