We introduce a longevity feature to the classical optimal dividend problem by adding a constraint on the time of ruin of the firm. We extend the results in [HJ15], now in context of one-sided Lévy risk models. We consider de Finettis problem in both scenarios with and without fix transaction costs, e.g. taxes. We also study the constrained analog to the so called Dual model. 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 indexed by Lagrange multipliers. Finally, we illustrate our findings with a series of numerical examples.

In this paper, we guarantee the existence and uniqueness (in the almost everywhere sense) of the solution to a Hamilton-Jacobi-Bellman (HJB) equation with gradient constraint and a partial integro-differential operator whose Lévy measure has bounded variation. This type of equation arises in a singular control problem, where the state process is a multidimensional jump-diffusion with jumps of finite variation and infinite activity. We verify, by means of ε-penalized controls, that the value function associated with this problem satisfies the aforementioned HJB equation.

The main goal of this paper is to establish existence, regularity and uniqueness results for the solution of a Hamilton-Jacobi-Bellman (HJB) equation, whose operator is an elliptic integro-differential operator. The HJB equation studied in this work arises in singular stochastic control problems where the state process is a controlled d-dimensional Lévy process.

We consider the bail-out optimal dividend problem under fixed transaction costs for a Lévy risk model with a constraint on the expected net present value of injected capital. In order to solve this problem, we first consider the bail-out optimal dividend problem under transaction costs and capital injection and show the optimality of reflected (c 1 , c 2 )-policies. We then find the optimal Lagrange multiplier, by showing that in the dual Laagrangian problem, the complementary slackness conditions are verified. Finally, we verify our results with some numerical examples.

We consider de Finetti's problem for spectrally one-sided Lévy risk models with control strategies that are absolutely continuous with respect to the Lebesgue measure. Furthermore, we consider the version with a constraint on the time of ruin. To characterize the solution to the aforementioned models, we first solve the optimal dividend problem with a terminal value at ruin and show the optimality of threshold strategies. Next, we introduce the dual Lagrangian problem and show that the complementary slackness conditions are satisfied, characterizing the optimal Lagrange multiplier. Finally, we illustrate our findings with a series of numerical examples.