An optimal dividend strategy in the discrete Sparre Andersen model with bounded dividend rates

• The optimal dividend-payment problem in discrete models has its own special significance.
• The bounded solution of an infinite set of discrete HJB equations is considered.
• Special transforms of value functions and Bellman’s recursive algorithm are applied.

We consider the discrete Sparre Andersen risk model and its derivative models by anew setting up the initial times, and assume that dividends are paid to the shareholders according to an admissible strategy with dividend rates bounded by a constant. The company controls the amount of dividends in order to maximize the cumulative expected discounted dividends prior to ruin. We show that the optimal value function is the unique bounded solution of a set of discrete Hamilton–Jacobi–Bellman equations. Moreover, we introduce Bellman’s recursive algorithm and offer a simpler algorithm to obtain the optimal strategy and the optimal value functions. Our method is mainly to transform the value functions. Numerical examples are presented to illustrate the transformation method.