Optimal Dividend Payment under a Ruin Constraint: Discrete Time and State Space

We consider optimal dividend payment under the constraint that the controlled risk process has a ruin probability which does not exceed a given bound. The underlying simple model has independent identically distributed total claims per year and a constant yearly premium, all integers. The solution to this constraint optimization problem is given in a modified Hamilton-Jacobi-Bellman (HJB) equation. It is shown that this equation has a solution, and a verification argument is given showing that the solution of the HJB equation is the value function of the optimization problem. The optimal dividend payment strategy is given in the usual feedback form.