The optimality of a control band policy

Consider a firm with an arbitrary profit function whose relative price follows a Brownian motion with negative drift. When the firm faces a fixed cost of price adjustment, we prove the optimal pricing policy is a control band if the following sufficient conditions are met: the profit function is continuous, strictly concave and single-peaked; moreover, together with its first and second derivatives, it is bounded in absolute value by a polynomial. We also demonstrate various ways of constructing the value function associated with the control band policy and show it has certain properties carried over from the profit function. Numerical examples are found to be consistent with empirical estimates regarding the frequency of price adjustments.