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We consider the optimal stopping problem consisting in, given a strong Markovprocess, a reward function and a discount rate, finding the stopping time suchthat the expected reward at the stopping time is maximum. The approach wefollow, has two main components: the Dynkin's characterization of the valuefunction as the smallest excessive function dominating the reward; and theRiesz representation of excessive functions in terms of the Green kernel, themain reference being Salminen 85. In the context of one-dimensional diffusionswe give a complete characterization of the solution under some assumptions onthe reward function. If the optimal stopping problem is a ray, we provide asimple equation to find the boundary and discuss the validity of the smooth fitprinciple. We include some new examples as the optimal stopping of the skewBrownian motion and the sticky Brownian motion. In particular, we considercases in which the smooth fit principle fails. In the general case, we proposean algorithm that finds the optimal stopping region when it is a disjoint unionof intervals. We also give a simple formula for the value function. Using thisalgorithm we solve some examples including polynomial rewards. For generalMarkov processes with continuous sample paths (for instance multidimensionaldiffusions) we provide a verification theorem and use it to solve a particularproblem. Finally we consider one-dimensional strong Markov processes with onlypositive (or only negative) jumps, and provide another verification theorem forright-sided (left-sided) problems. As applications of our results we addressthe problem of pricing an American put option in a L\'evy market, and alsosolve an optimal stopping problem for a L\'evy driven Ornstein-Uhlenbeckprocess.

Optimal Stopping for Strong Markov Processes: Explicit solutions and  verification theorems for diffusions, multidimensional diffusions, and  jump-processes