Given a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in\mathbb{T}},\mathbb{P})$ , a process $(X_t)_{t\in\mathbb{T}}$ is a martingale if it satisfies the equality \begin{equation*} \mathbb{E}[X_t\mid\mathcal{F}_s]=X_s \end{equation*}for all $s<t$ in the index set $\mathbb{T}$ . Doob's optional sampling theorem says that this equality still holds if the times $s,t$ are replaced by bounded stopping times $S,T$ . In this case, the $\sigma$ -algebra $\mathcal{F}_s$ is replaced by the collection of events observable at the random time $S$ , \begin{equation*} \mathcal{F}_S=\left\{A\in\mathcal{F}:A\cap\{S\le t\}\in\mathcal{F}_t\textrm{ for all }t\in\mathbb{T}\right\}. \end{equation*}In discrete-time, when the index set $\mathbb{T}$ is countable, the result is as follows.

This theorem shows, amongst other things, that in the case of a fair casino, where your return is a martingale, betting strategies involving `knowing when to quit' do not enhance your expected return.

In continuous-time, when the index set $\mathbb{T}$ an interval of the real numbers, then the stopping times $S,T$ can have a continuous distribution and $X_S,X_T$ need not be measurable quantities. Then, it is necessary to place conditions on the sample paths of the process $X$ . In particular, Doob's optional sampling theorem holds in continuous-time if $X$ is assumed to be right-continuous.