  We propose a recursive algorithm for the numerical computation of the optimal
value function $\inf_{t\le\tau\le T} E \Big[\sup_{0\le s\le T } Y_s / Y_{\tau}
\big| {\cal F}_t\Big]$ over the stopping times $\tau$ with respect to the
filtration of a geometric Brownian motion $Y_t$ with Markovian regime
switching. This method allows us to determine the boundary functions of the
optimal stopping set when no associated Volterra integral equation is
available. It applies in particular when regime-switching drifts have mixed
signs, in which case the boundary functions may not be monotone.
