Optimal Stopping with Randomly Arriving Opportunities to Stop

Abstract: We develop methods to solve general optimal stopping problems with opportunities to stop that arrive randomly. Such problems occur naturally in applications with market frictions. Pivotal to our approach is that our methods operate on random rather than deterministic time scales. This enables us to convert the original problem into an equivalent discrete-time optimal stopping problem with $\mathbb{N}_{0}$-valued stopping times and a possibly infinite horizon. To numerically solve this problem, we design a random times least squares Monte Carlo method. We also analyze an iterative policy improvement procedure in this setting. We illustrate the efficiency of our methods and the relevance of randomly arriving opportunities in a few examples.