  We study a simple singular control problem for a Brownian motion with
constant drift and variance reflected at the origin. Exerting control pushes
the process towards the origin and generates a concave increasing
state-dependent yield which is discounted at a fixed rate. The most interesting
feature of the problem is that its solution can be more complicated than
anticipated. Indeed, for some parameter values, the optimal policy involves two
reflecting barriers and one repelling boundary, the action region being the
union of two disjoint intervals. We also show that the apparent anomaly can be
understood as involving a switch between two strategies with different risk
profiles: The risk-neutral decision maker initially gambles on the more risky
strategy, but lowers risk if this strategy underperforms.
