A Characterization of Stopping Times

Abstract

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 $\mathscr{F}_t$.