Doob’s optional sampling theorem

Given a filtered probability space (Ω,,(t)t𝕋,), a process (Xt)t𝕋 is a martingaleMathworldPlanetmath if it satisfies the equality


for all s<t in the index setMathworldPlanetmathPlanetmath 𝕋. Doob’s optional sampling theoremMathworldPlanetmath says that this equality still holds if the times s,t are replaced by bounded stopping times S,T. In this case, the σ-algebra s is replaced by the collectionMathworldPlanetmath of events observable at the random time S (,

S={A:A{St}t for all t𝕋}.

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

Doob’s Optional Sampling Theorem.

Suppose that the index set T is countable and that ST are stopping times bounded above by some constant cT. If (Xt) is a martingale then XT is an integrable random variableMathworldPlanetmath and

𝔼[XT|S]=XS, almost surely. (1)

Similarly, if X is a submartingale then XT is integrable and

𝔼[XT|S]XS, almost surely. (2)

If X is a supermartingale then XT is integrable and

𝔼[XT|S]XS, 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 XS,XT 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