OPTIMAL INVESTMENT POLICY AND DIVIDEND PAYMENT STRATEGY IN AN INSURANCE COMPANY

We consider in this paper the optimal dividend problem for an insurance company whose uncontrolled reserve process evolves as a classical Cramer-Lundberg process. The firm has the option of investing part of the surplus in a Black-Scholes financial market. The objective is to find a strategy consisting of both investment and dividend payment policies which maximizes the cumulative expected discounted dividend pay-outs until the time of bankruptcy. We show that the optimal value function is the smallest viscosity solution of the associated second-order integro-differential Hamilton-Jacobi-Bellman equation. We study the regularity of the optimal value function. We show that the optimal dividend payment strategy has a band structure. We find a method to construct a candidate solution and obtain a verification result to check optimality. Finally, we give an example where the optimal dividend strategy is not barrier and the optimal value function is not twice continuously differentiable.