Does anyone have a good explanation of the name, and why Doob chose it? It states the following: if $T$ is a stopping time such that $\mathbb{P}(T < \infty)$, and $M_n$ is a uniformly integrable martingale, then $$ E[M_T] = E[M_0]. $$
1
-
$\begingroup$ A "stopping time" is alternately known as an "optional time". $\endgroup$– Gerald EdgarCommented Feb 13, 2017 at 15:26
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
0
"Optional stopping" and "optional sampling" refer to a strategy of peeking while you are sampling and then, based on what you find, exercise the option to quit sampling.
Doob's theorem states that in a fair casino, where your return is a martingale, you cannot increase your expected return if you are given the option to stop betting (= sampling) when you think "it's time to quit".
The names originate from J. L. Doob, Stochastic Processes (Wiley, 1953): page 300, 366. Doob uses "optional sampling" for the stochastic process itself, and "optional stopping" for the random variable that represents the stopping time.
answered Feb 13, 2017 at 15:02