Optimal dividend payment under a ruin constraint: discrete time and state space

A modified Hamilton-Jacobi-Bellman (HJB) equation is derived for the problem of optimal dividend payment under a ruin constraint, for discrete time and state space. This equation has a classical solution, and a verification argument is given which shows that the solution is the value function of the problem, and that the maximizer in the HJB equation defines the optimal dividend payment strategy in feedback form.