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

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.