Optimal investment policy and dividend payment strategy in an insurance company

摘要

We consider in this paper the optimal dividend problem for an insurancecompany whose uncontrolled reserve process evolves as a classicalCram\'{e}r--Lundberg process. The firm has the option of investing part of thesurplus in a Black--Scholes financial market. The objective is to find astrategy consisting of both investment and dividend payment policies whichmaximizes the cumulative expected discounted dividend pay-outs until the timeof bankruptcy. We show that the optimal value function is the smallestviscosity solution of the associated second-order integro-differentialHamilton--Jacobi--Bellman equation. We study the regularity of the optimalvalue function. We show that the optimal dividend payment strategy has a bandstructure. We find a method to construct a candidate solution and obtain averification result to check optimality. Finally, we give an example where theoptimal dividend strategy is not barrier and the optimal value function is nottwice continuously differentiable.