The Expected Value of an Everywhere Stopped Martingale

Abstract

If the coordinate random variables $\{X_t\}$ on either $C\lbrack 0, \infty)$ or $D\lbrack 0, \infty)$ form a martingale, then for every stopping time $\tau$ which is everywhere finite, $E(X_\tau)$, if defined, equals $E(X_0)$. This version of the optional sampling theorem is not covered by Doob's classical result [1].