  This paper studies a class of zero-sum stopping game in a regime switching
model. A verification theorem as a sufficient criterion for Nash equilibriums
is established based on a set of variational inequalities (VIs). Under an
appropriate regularity condition for solutions to the VIs, a suitable system of
algebraic equations is derived via the so-called smooth-fit principle. Explicit
Nash equilibrium stopping rules of threshold-type for the two players and the
corresponding value function of the game in closed form are obtained. Numerical
experiments are reported to demonstrate the dependence of the threshold levels
on various model parameters. A reduction to the case with no regime switching
is also presented as a comparison.
