  We consider the optimal stopping problem consisting in, given a strong Markov
process, a reward function and a discount rate, finding the stopping time such
that the expected reward at the stopping time is maximum. The approach we
follow, has two main components: the Dynkin's characterization of the value
function as the smallest excessive function dominating the reward; and the
Riesz representation of excessive functions in terms of the Green kernel, the
main reference being Salminen 85. In the context of one-dimensional diffusions
we give a complete characterization of the solution under some assumptions on
the reward function. If the optimal stopping problem is a ray, we provide a
simple equation to find the boundary and discuss the validity of the smooth fit
principle. We include some new examples as the optimal stopping of the skew
Brownian motion and the sticky Brownian motion. In particular, we consider
cases in which the smooth fit principle fails. In the general case, we propose
an algorithm that finds the optimal stopping region when it is a disjoint union
of intervals. We also give a simple formula for the value function. Using this
algorithm we solve some examples including polynomial rewards. For general
Markov processes with continuous sample paths (for instance multidimensional
diffusions) we provide a verification theorem and use it to solve a particular
problem. Finally we consider one-dimensional strong Markov processes with only
positive (or only negative) jumps, and provide another verification theorem for
right-sided (left-sided) problems. As applications of our results we address
the problem of pricing an American put option in a L\'evy market, and also
solve an optimal stopping problem for a L\'evy driven Ornstein-Uhlenbeck
process.
