# Doob’s optional sampling theorem

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

 $\mathbb{E}[X_{t}\mid\mathcal{F}_{s}]=X_{s}$

for all $s 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 $S$ (http://planetmath.org/SigmaAlgebraAtAStoppingTime),

 $\mathcal{F}_{S}=\left\{A\in\mathcal{F}:A\cap\{S\leq t\}\in\mathcal{F}_{t}% \textrm{ for all }t\in\mathbb{T}\right\}.$

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

###### Doob’s Optional Sampling Theorem.

Suppose that the index set $\mathbb{T}$ is countable and that $S\leq T$ are stopping times bounded above by some constant $c\in\mathbb{T}$. If $(X_{t})$ is a martingale then $X_{T}$ is an integrable random variable  and

 $\mathbb{E}[X_{T}|\mathcal{F}_{S}]=X_{S},\ \mathbb{P}\textrm{ almost surely}.$ (1)

Similarly, if $X$ is a submartingale then $X_{T}$ is integrable and

 $\mathbb{E}[X_{T}|\mathcal{F}_{S}]\geq X_{S},\ \mathbb{P}\textrm{ almost surely}.$ (2)

If $X$ is a supermartingale then $X_{T}$ is integrable and

 $\mathbb{E}[X_{T}|\mathcal{F}_{S}]\leq X_{S},\ \mathbb{P}\textrm{ almost surely}.$ (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 $\mathbb{T}$ 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.

Title Doob’s optional sampling theorem DoobsOptionalSamplingTheorem 2013-03-22 16:43:41 2013-03-22 16:43:41 skubeedooo (5401) skubeedooo (5401) 8 skubeedooo (5401) Theorem msc 60G44 msc 60G46 msc 60G42 Martingale StoppingTime SigmaAlgebraAtAStoppingTime