A note on the optimal control of a cash balance problem

This paper provides an analytical solution to a cash management problem when cash income and demand are described by Compound Poisson processes. The paper generalizes past results in the cash management literature to arbitrary income and demand distribution functions. Further, our results can be applied as well in the area of banking. Throughout the paper we restrict attention to the family of control barrier policies. These consist in hedging cash up to a critical level and investing all incoming cash exceeding this level. We employ a long-run average cost criterion to determine an optimal control barrier. A diffusion approximation of the cash level process (income less demand) is used to obtain a simpler expression for the average cost and to yield a closed form solution to the optimal control barrier. For demonstration purposes, an example is resolved.