Optimal dividend payments under a time of ruin constraint: Exponential claims

We consider the optimal dividends problem under the Cramér-Lundberg model with exponential claim sizes subject to a constraint on the expected 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 can be obtained as the point-wise infimum of auxiliary value functions indexed by Lagrange multipliers. We also present a series of numerical examples.