Solutions for Poissonian stopping problems of linear diffusions via extremal processes

We develop a general yet simple technique for solving Poissonian timing problems of linear diffusions by relying on the close connection of the extremal processes and the first passage times of the underlying diffusion. We provide a closed-form representation of the expected value gained by employing an ordinary first passage time-based stopping strategy. This approach simplifies the determination of the optimal policy, transforming it into an analysis of ordinary first-order optimality conditions. We relate our findings to various existing approaches for solving stopping problems of linear diffusions and express the optimality conditions in a single boundary setting in a form familiar from optimal stopping of Lévy-processes.