Some Results on Optimal Dividend Problem in Two Risk Models

The compound Poisson risk model and the compound Poisson risk model perturbed by diffusion are considered in the presence of a dividend barrier with solvency constraints. Moreover, it extends the known result due to [1]. Ref. [1] finds the optimal dividend policy is of a barrier type for a jump-diffusion model with exponentially distributed jumps. In this paper, it turns out that there can be two different solutions depending on the model’s parameters. Furthermore, an interesting result is given: the proportional transaction cost has no effect on the dividend barrier. The objective of the corporation is to maximize the cumulative expected discounted dividends payout with solvency constraints before the time of ruin. It is well known that under some reasonable assumptions, optimal dividend strategy is a barrier strategy, i.e., there is a level b_{1}(b_{2}) so that whenever surplus goes above the level b_{1}(b_{2}), the excess is paid out as dividends. However, the optimal level b_{1}(b_{2}) may be unacceptably low from a solvency point of view. Therefore, some constraints should imposed on an insurance company such as to pay out dividends unless the surplus has reached a level b^{1}_{c}>b_{1}(b^2_{c}>b_{2}) . We show that in this case a barrier strategy at b^{1}_{c}(b^2_{c}) is optimal.