  We study an optimal stopping problem when the state process is governed by a
general Feller process. In particular, we examine viscosity properties of the
associated value function with no a priori assumption on the stochastic
differential equation satisfied by the state process. Our approach relies on
properties of the Feller semigroup. We present conditions on the state process
under which the value function is the unique viscosity solution to an
Hamilton-Jacobi-Bellman (HJB) equation associated with a particular operator.
More specifically, assuming that the state process is a Feller process, we
prove uniqueness of the viscosity solution which was conjectured in [26]. We
then apply our results to study viscosity property of optimal stopping problems
for some particular Feller processes, namely diffusion processes with piecewise
coefficients and semi-Markov processes. Finally, we obtain explicit value
functions for optimal stopping of straddle options, when the state process is a
reflected Brownian motion, Brownian motion with jump at boundary and regime
switching Feller diffusion, respectively (see Section 8).
