Optimal Control Strategy For Dividend Payments And Capital Injections In The Discrete Sparre Andersen Risk Model With Penalties For Deficits

We consider the discrete Sparre Andersen risk model with capital injections and dividend payments. The company controls the amount of dividends paid to the shareholders as well as the capital injections in order to maximize the cumulative expected discounted dividends minus the discounted capital injections and the discounted penalties for deficits prior to ruin. We show that the optimal value function is the unique bounded solution of a set of discrete HJB equations,and the optimal control strategy is a two-barriers strategy given the distribution of the first claim inter-occurrence time of the delayed renewal process. We obtain some properties of the optimal strategy and the optimal condition for ruining the company. Our method is mainly to transform the value functions. We offer an efficient algorithm for solving the corresponding image function to obtain the optimal strategy and optimal value function. Numerical results are provided to illustrate the algorithm and the impact of the penalties.