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].
Citation
S. Ramakrishnan. W. D. Sudderth. "The Expected Value of an Everywhere Stopped Martingale." Ann. Probab. 14 (3) 1075 - 1079, July, 1986. https://doi.org/10.1214/aop/1176992461
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