Chapter 29

Martingales29.1 Conditional Expectation

From Observation 27.4.5 on Page 748, it was shown that the conditional expectation of arandom variableX given some others really is just what the words suggest. Given ω ∈Ω,it results in a value for the “other” random variables and then you essentially take theexpectation ofX given this information which yields the value of the conditional expecta-tion of X given the other random variables. It was also shown in Lemma 27.4.4 that thisgives the same result as finding a σ (X1, · · · ,Xn) measurable function Z such that for allF ∈ σ (X1, · · · ,Xn) , ∫

FXdP =

∫FZdP

This was done for a particular type of σ algebra but there is no need to be this specialized.The following is the general version of conditional expectation given a σ algebra. It makesperfect sense to ask for the conditional expectation given a σ algebra and this is what willbe done from now on.

Definition 29.1.1 Let (Ω,M ,P) be a probability space and let S ⊆ F be twoσ algebras contained in M . Let f be F measurable and in L1 (Ω;W ) for W a Banachspace. Then E ( f |S ) , called the conditional expectation of f with respect to S is definedas follows:

E ( f |S ) is S measurable

For all E ∈S , ∫E

E ( f |S )dP =∫

Ef dP

The existence and uniqueness of the conditional expectation is described earlier in The-orem 24.12.1 on Page 702. For convenience, here is this theorem.

Theorem 29.1.2 Let E be a separable Banach space and X ∈ L1 (Ω;E,M ) whereX is measurable with respect to M and let S be a σ algebra which is contained in M .Then there exists a unique Z ∈ L1 (Ω;E,S ) such that for all A ∈S ,∫

AXdP =

∫A

ZdP

Denoting this Z as E (X |S ) , it follows

∥E (X |S )∥ ≤ E (∥X∥ |S ) .

A few properties are described next. Let W be a separable Banach space in the follow-ing lemma.

Lemma 29.1.3 The above is well defined. Also, if S ⊆F then if X ∈ L1 (Ω;W ) ,

E (X |S ) = E (E (X |F ) |S ) . (29.1)

If Z is in L∞ (Ω;W ′)bounded and measurable in S then

ZE (X |S ) = E (ZX |S ) . (29.2)

781