  We study the regularity of the stochastic representation of the solution of a
class of initial-boundary value problems related to a regime-switching
diffusion. This representation is related to the value function of a
finite-horizon optimal stopping problem such as the price of an American-style
option in finance. We show continuity and smoothness of the value function
using coupling and time-change techniques. As an application, we find the
minimal payoff scenario for the holder of an American-style option in the
presence of regime-switching uncertainty under the assumption that the
transition rates are known to lie within level-dependent compact sets.
