Abstract: This paper deals with the optimal dividend payment and  capital injection problem for a two-dimensional compound Poisson risk model which constructs correlation among the two claims. The objective of the corporation is to maximize the discounted dividend payments minus the penalized discounted capital injections. The problem is formulated as a stochastic control problem. By solving the corresponding Hamilton-Jacobi-Bellman (HJB) equation, we obtain the optimal dividend strategy of the problem. We solve this problem explicitly in the case of exponential claim amount distributions.

Key words: optimal dividends, capital injection, Hamilton-Jacobi-Bellman (HJB) equation, stochastic control, two-dimensional compound Poisson risk model