A Note on the Optimal Dividend Payments for the Jump-Diffusion Process with Solvency Constraints

This paper extends the known result due to Belhaj who found the optimal dividend policy is of a barrier type for a jump-diffusion model with exponentially distributed jumps. It turns out that there can be essentially two different solutions depending on the model´s parameters. It also deals with the optimal control problem for the jump-diffusion process with solvency constraints. The objective of the corporation is to maximize the cumulative expected discounted dividends payout with solvency constraints. It is well known that under some reasonable assumptions, optimal dividend strategy is a barrier strategy, i.e., there is a level b* so that whenever surplus goes above b*, the excess is paid out as dividends. However, the optimal level b* 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 b₀ >; b*. We show that in this case a barrier strategy at b0 is optimal.