Optimal dividend payouts for diffusions with solvency constraints

We consider a company where surplus follows a diffusion process and whose objective is to maximize expected discounted dividend payouts to the shareholders. It is well known that under some reasonable assumptions, optimality is achieved by using a barrier strategy, i.e. there is a level b* so that whenever suplus goes above b*, the excess is paid out as dividends. However, the optimal level b* may be unaccaptably low, and the company may be prohibited, either by internal clauses or by external reasons such as solvency restrictions imposed on an insurance company, to pay out dividends unless the surplus has reached a level b(0) > b*. We show that in this case a barrier strategy at b(0) is optimal. Finally, it is discussed how the barrier b(0) can be determined, and we suggest to use arguments from risk theory. More concretely, we let b(0) be the smallest barrier so that the probability that the surplus will be negative within a time horizon T is not larger than some E when initial surplus equals b(0). It is shown theoretically how b(0) can be calculated using this method, and examples are given for two special cases.