Optimal Singular Dividend Problem under the Sparre Anderson Model

Consider an insurance company for which the reserve process follows the Sparre Anderson model. In this paper, we study the optimal dividend problem for such a company as Bai, Ma and Xing [9] do. However, we remove the 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 HJB equation is a variational inequality involving a first order integro-differential operator and a gradient constraint. We use other techniques to prove the regularity properties for the value function and show that the value function is a constrained viscosity solution of the associated HJB equation. In addition, we show that the value function is the upper semi-continuous envelop of the supremum for a class of subsolutions.