Consider the problem of a central bank that wants to manage the exchange

rate between its domestic currency and a foreign one. The central bank can

purchase and sell the foreign currency, and each intervention on the exchange market

leads to a proportional cost whose instantaneous marginal value depends on the

current level of the exchange rate. The central bank aims at minimizing the total

expected costs of interventions on the exchange market, plus a total expected holding

cost. We formulate this problem as an infinite time-horizon stochastic control problem

with controls that have paths which are locally of bounded variation. The exchange

rate evolves as a general linearly controlled one-dimensional diffusion, and the two

nondecreasing processes giving the minimal decomposition of a bounded-variation

control model the cumulative amount of foreign currency that has been purchased

and sold by the central bank. We provide a complete solution to this problem by

finding the explicit expression of the value function and a complete characterization

of the optimal control. At each instant of time, the optimally controlled exchange rate

is kept within a band whose size is endogenously determined as part of the solution to

the problem. We also study the expected exit time from the band, and the sensitivity

of the width of the band with respect to the model's parameters in the case when

the exchange rate evolves (in absence of any intervention) as an Ornstein-Uhlenbeck

process, and the marginal costs of controls are constant. The techniques employed in

the paper are those of the theory of singular stochastic control and of one-dimensional diffusions