A diffusion model for optimal dividend distribution for a company with constraints on risk control

This paper investigates a model of a corporation which faces constant liability payments and which can choose a production/business policy from an available set of control policies with different expected profits and risks. The objective is to find a business policy and a dividend distribution scheme so as to maximize the expected present value of the total dividend distributions. The main feature of this paper is that there are constraints on business activities such as inability to completely eliminate risk ( even at the expense of reducing the potential profit to zero) or when such a risk cannot exceed a certain level. The case in which there is no restriction on the dividend pay-out rates is dealt with. This gives rise to a mixed regular-singular stochastic control problem. First the value function is analyzed in great detail and in particular is shown to be a viscosity solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation. Based on this it is further proved that the value function must be twice continuously differentiable. Then a delicate analysis is carried out on the HJB equation, leading to an explicit expression of the value function as well as the optimal policies.