Let $X$ be a semimartingale, and $S$ its Snell envelope. Under the assumption that $X$ and $S$ are continuous semimartingales in $H^1$, this article obtains a new, maximal, characterisation of $S$, and gives an application to the optimal stopping of functions of diffusions. We present a counterexample to the standard assertion that $S$ is just "a martingale on the go-region and $X$ on the stop-region."
"Local Times, Optimal Stopping and Semimartingales." Ann. Probab. 21 (1) 329 - 339, January, 1993. https://doi.org/10.1214/aop/1176989407