Doob’s optional sampling theorem

Given a filtered probability space $(\mathrm{\Omega },ℱ,{({ℱ}_t)}_{t\in 𝕋},\mathbb{P})$, a process ${(X_t)}_{t\in 𝕋}$ is a martingale if it satisfies the equality

$$𝔼[X_t\mid {ℱ}_s]=X_s$$

for all in the index set $𝕋$. 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 ${ℱ}_s$ is replaced by the collection of events observable at the random time $S$ (http://planetmath.org/SigmaAlgebraAtAStoppingTime),

$${ℱ}_S=\{A\in ℱ:A\cap \{S\le t\}\in {ℱ}_t\text{ for all }t\in 𝕋\}.$$

In discrete-time, when the index set $𝕋$ is countable, the result is as follows.

Doob’s Optional Sampling Theorem.

Suppose that the index set $\mathrm{T}$ is countable and that $S\mathrm{\le }T$ are stopping times bounded above by some constant $c\mathrm{\in }\mathrm{T}$.If $\mathrm{(}{X}_t\mathrm{)}$ is a martingale then $X_T$ is an integrable random variable and

$$𝔼[X_T|{ℱ}_S]=X_S,\mathbb{P}\mathit{\text{ almost surely}}.$$(1)

Similarly, if $X$ is a submartingale then $X_T$ is integrable and

$$𝔼[X_T|{ℱ}_S]\ge X_S,\mathbb{P}\mathit{\text{ almost surely}}.$$(2)

If $X$ is a supermartingale then $X_T$ is integrable and

$$𝔼[X_T|{ℱ}_S]\le X_S,\mathbb{P}\mathit{\text{ almost surely}}.$$(3)

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 $𝕋$ 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.