30.3. FILTRATIONS 823

This is called the predictable σ algebra. and the sets in this σ algebra are called thepredictable sets. Denote by PT the intersection of P∞ to [0,T ]×Ω. A stochastic processX which maps either [0,T ]×Ω or [0,∞)×Ω to E, a separable real Banach space is calledpredictable if for every Borel set A ∈B (E) , it follows X−1 (A) ∈PT or P∞.

This is a lot like product measure except one of the σ algebras is changing.

Proposition 30.3.12 Let Ft be a filtration as above and let X be a predictable stochas-tic process. Then X is Ft adapted.

Proof: Let s0 > 0 and define

Gs0 ≡{

S ∈P∞ : Ss0 ∈Fs0

}where

Ss0 ≡ {ω ∈Ω : (s0,ω) ∈ S} .

ΩSs0

s0

It is clear Gs0 is a σ algebra. The next step is to show Gs0 contains the sets

(s, t]×F,F ∈Fs (30.19)

and{0}×F, F ∈F0. (30.20)

It is clear {0}×F is contained in Gs0 because ({0}×F)s0= /0 ∈Fs0 . Similarly, if s ≥ s0

or if s, t < s0 then ((s, t]×F)s0= /0 ∈Fs0 . The only case left is for s < s0 and t ≥ s0. In this

case, letting As ∈Fs, ((s, t]×As)s0= As ∈Fs ⊆Fs0 . Therefore, Gs0 contains all the sets

of the form given in 30.19 and 30.20 and so since P∞ is the smallest σ algebra containingthese sets, it follows P∞ = Gs0 . The case where s0 = 0 is entirely similar but shorter.

Therefore, if X is predictable, letting A ∈B (E) , X−1 (A) ∈P∞ or PT and so(X−1 (A)

)s ≡ {ω ∈Ω : X (s,ω) ∈ A}= X (s)−1 (A) ∈Fs

showing X (t) is Ft adapted. ■Another way to see this is to recall the progressively measurable functions are adapted.

Then show the predictable sets are progressively measurable.

Proposition 30.3.13 Let P denote the predictable σ algebra and let R denote theprogressively measurable σ algebra. Then P ⊆R.

Proof: Let G denote those sets of P such that they are also in R. Then G clearlycontains the π system of sets {0}×A,A ∈F0, and (s, t]×A,A ∈Fs. Furthermore, G isclosed with respect to countable disjoint unions and complements. It follows G containsthe σ algebra generated by this π systems which is P . ■