  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.
