  This paper studies a finite-fuel two-dimensional degenerate singular
stochastic control problem under regime switching that is motivated by the
optimal irreversible extraction problem of an exhaustible commodity. A company
extracts a natural resource from a reserve with finite capacity, and sells it
in the market at a spot price that evolves according to a Brownian motion with
volatility modulated by a two-state Markov chain. In this setting, the company
aims at finding the extraction rule that maximizes its expected discounted cash
flow, net of the costs of extraction and maintenance of the reserve. We provide
expressions both for the value function and for the optimal control. On the one
hand, if the running cost for the maintenance of the reserve is a convex
function of the reserve level, the optimal extraction rule prescribes a
Skorokhod reflection of the (optimally) controlled state process at a certain
state and price dependent threshold. On the other hand, in presence of a
concave running cost function it is optimal to instantaneously deplete the
reserve at the time at which the commodity's price exceeds an endogenously
determined critical level. In both cases, the threshold triggering the optimal
control is given in terms of the optimal stopping boundary of an auxiliary
family of perpetual optimal selling problems with regime switching.
