# Doob’s optional sampling theorem

Given a filtered probability space $(\mathrm{\Omega},\mathcal{F},{({\mathcal{F}}_{t})}_{t\in \mathbb{T}},\mathbb{P})$, 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 $$ 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),

$${\mathcal{F}}_{S}=\{A\in \mathcal{F}:A\cap \{S\le t\}\in {\mathcal{F}}_{t}\text{for all}t\in \mathbb{T}\}.$$ |

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 $\mathrm{T}$ 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

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

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

$$\mathbb{E}[{X}_{T}|{\mathcal{F}}_{S}]\ge {X}_{S},\mathbb{P}\mathit{\text{almost surely}}.$$ | (2) |

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

$$\mathbb{E}[{X}_{T}|{\mathcal{F}}_{S}]\le {X}_{S},\mathbb{P}\mathit{\text{almost surely}}.$$ | (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 $\mathbb{T}$ 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 |
---|---|

Canonical name | DoobsOptionalSamplingTheorem |

Date of creation | 2013-03-22 16:43:41 |

Last modified on | 2013-03-22 16:43:41 |

Owner | skubeedooo (5401) |

Last modified by | skubeedooo (5401) |

Numerical id | 8 |

Author | skubeedooo (5401) |

Entry type | Theorem |

Classification | msc 60G44 |

Classification | msc 60G46 |

Classification | msc 60G42 |

Related topic | Martingale |

Related topic | StoppingTime |

Related topic | SigmaAlgebraAtAStoppingTime |