Ferrari G, Vargiolu T. On the Singular Control of Exchange Rates . Center for Mathematical Economics Working Papers. Vol 594. Bielefeld: Center for Mathematical Economics; 2017.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 diffusion
