818 CHAPTER 30. CONTINUOUS STOCHASTIC PROCESSES

the completion of the smallest σ algebra such that each X (s) is measurable for all s ≤ t.This gives an example of a filtration to which X (t) is adapted which satisfies 1. Moregenerally, suppose X (t) is adapted to a filtration, Gt . Define

Ft ≡ ∩s>tGs

ThenFt+ ≡ ∩s>tFs = ∩s>t ∩r>s Gr = ∩s>tFs ≡Ft .

and each X (t) is measurable with respect to Ft . Thus there is no harm in assuming astochastic process adapted to a filtration can be modified so the filtration is normal. Alsonote that Ft defined this way will be complete so if A∈Ft has P(A) = 0 and if B⊆ A, thenB ∈Ft also. This is because this relation between the sets and the probability of A beingzero, holds for this pair of sets when considered as elements of each Gs for s > t. HenceB ∈ Gs for each s > t and is therefore one of the sets in Ft .

What is the description of a progressively measurable set?

t

QQ⋂[0, t]×Ω

It means that for Q progressively measurable, Q∩ [0, t]×Ω as shown in the abovepicture is B ([0, t])×Ft measurable. It is like saying a little more descriptively that thefunction is progressively product measurable.

I shall generally assume the filtration is normal.

Observation 30.3.3 If X is progressively measurable, then it is adapted. Furthermorethe progressively measurable sets, those E∩ [0,T ]×Ω for which XE is progressively mea-surable form a σ algebra.

To see why this is, consider X progressively measurable and fix t. Then (s,ω) →X (s,ω) for (s,ω)∈ [0, t]×Ω is given to be B ([0, t])×Ft measurable, the ordinary productmeasure and so fixing any s∈ [0, t] , it follows the resulting function of ω is Ft measurable.In particular, this is true upon fixing s = t. Thus ω→ X (t,ω) is Ft measurable and so X (t)is adapted.

A set E ⊆ [0,T ]×Ω is progressively measurable means that XE is progressively mea-surable. That is XE restricted to [0, t]×Ω is B ([0, t])×Ft measurable. In other words, Eis progressively measurable if

E ∩ ([0, t]×Ω) ∈B ([0, t])×Ft .

If Ei is progressively measurable, does it follow that E ≡∪∞i=1Ei is also progressively mea-

surable? Yes.

E ∩ ([0, t]×Ω) = ∪∞i=1Ei∩ ([0, t]×Ω) ∈B ([0, t])×Ft