Optimal Singular Dividend Problem Under the Sparre Andersen Model

摘要

In this paper, we study the optimal dividend problem assuming that the underlying reserve process follows the Sparre Andersen model. In this model, there is no constant restriction on the dividend rates, i.e., the optimization problem is of singular type. In this case, the value function is no longer bounded and the associated Hamilton-Jacobi-Bellman equation is a variational inequality involving a first-order integro-differential operator and a gradient constraint. We prove the regularity properties for the value function by constructing strategies and show that the value function is a constrained viscosity solution of the associated Hamilton-Jacobi-Bellman equation. In addition, we prove that the value function is the upper semicontinuous envelope of the supremum for a class of subsolutions.