In this paper the authors study a model for the optimal operation of a bank or insurance company which was recently introduced by Peura and Keppo. The model generalizes a previous one of Milne and Robertson by allowing the bank to raise capital as well as to pay out dividends. Optimal operation of the bank is determined by solving an optimal control problem. In this paper it is shown that the solution of the optimal control problem proposed by Peura and Keppo exists for all values of the parameters and is unique.