Optimal dividend payout under compound Poisson income

We consider an individual who receives income, which may be either positive or negative, and is allowed to pay out a dividend at any time as long as the accumulated income remains positive. In case the accumulated income become negative at some point in time, the individual declares bankruptcy, pays a penalty based on his accumulated income, and the process stops. Assuming that the input process is described by a compound Poisson process and that the individual's value is given by the accumulated dividends minus the penalty, both appropriately discounted, we demonstrate an optimal policy for paying dividends and provide an iterative means for estimating the corresponding value.