Let $R$ be a random time in $\mathscr{F}_\infty$, the terminal element of a filtration $\mathscr{F}_t$ satisfying the usual hypotheses. It is shown that if optimal sampling holds at $R$ for all bounded martingales, then $R$ is optional. If $\mathscr{F}_t$ is the natural pseudo-path filtration of a measurable process $X_t$, then $R$ is optional if (and only if) the conditional distribution of $X_{R + .}$ given $\mathscr{F}_R$ is $Z_R$, where $Z_t$ is an optional version of the conditional distribution of $X_{t +.}$ given $\mathscr{F}_t$.
@article{1176988615,
author = {Knight, Frank B. and Maisonneuve, Bernard},
title = {A Characterization of Stopping Times},
journal = {Ann. Probab.},
volume = {22},
number = {4},
year = {1994},
pages = { 1600-1606},
language = {en},
url = {http://dml.mathdoc.fr/item/1176988615}
}
Knight, Frank B.; Maisonneuve, Bernard. A Characterization of Stopping Times. Ann. Probab., Tome 22 (1994) no. 4, pp. 1600-1606. http://gdmltest.u-ga.fr/item/1176988615/