Before revision:  This problem is studied with dynamic programming techniques and with regularization arguments . Using results from the recent literature we obtain the existence of classical solutions to the dynamic programming equation in a regularized version of the model. From this the optimal strategy in the regularized model is straightforward to compute. We give convergence results which show that this strategy is epsilon-optimal in the original model .
After revision:  This problem is studied with dynamic programming techniques and with regularization arguments .