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Consider an insurance company for which the reserve process follows theSparre Anderson model. In this paper, we study the optimal dividend problem forsuch a company as Bai, Ma and Xing [9] do. However, we remove the constantrestriction on the dividend rates, i.e. the optimization problem is of singulartype. In this case, the value function is no longer bounded and the associatedHJB equation is a variational inequality involving a first orderintegro-differential operator and a gradient constraint. We use othertechniques to prove the regularity properties for the value function and showthat the value function is a constrained viscosity solution of the associatedHJB equation. In addition, we show that the value function is the uppersemi-continuous envelop of the supremum for a class of subsolutions.

Optimal Singular Dividend Problem under the Sparre Anderson Model