2542 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS

It will be assumed for the sake of convenience that p≥ 2. It follows that

V ⊆W , W ′ ⊆ V ′

The entire presentation will be based on the following lemma.

Lemma 76.3.1 Let V ≡ Lp ([0,T ]×Ω,V ) where V is a separable Banach space and theσ algebra of measurable sets consists of those which are progressively measurable. Thenfor h ∈ (0,T ) , τh : V → V .

Proof: First consider Q which is a progressively measurable set. Is it the case thatτhXQ is also progressively measurable? Define Q+h as

Q+h≡ {(t +h,ω) : (t,ω) ∈ Q}

Then

τhXQ (t,ω) =

{XQ+h (t,ω) if t ≥ h0 if t < h

Is this function progressively measurable? For (s,ω) ∈ [0, t]×Ω, we have the following

0 < α ≤ 1, [(s,ω) : τhXQ (s,ω)≥ α] = [h, t]×Ω∩ (Q+h)

α > 1, [(s,ω) : τhXQ (s,ω)≥ α] = /0 ∈B ([0, t])×Ft

α ≤ 0, [(s,ω) : τhXQ (s,ω)≥ α] = [0, t]×Ω ∈B ([0, t])×Ft

It suffices to show that for t ≥ h, [h, t]×Ω∩ (Q+h) is B ([0, t])×Ft measurable. It isknown that [0, t]×Ω∩Q is B ([0, t])×Ft measurable and also that [0, t−h]×Ω∩Q isB ([0, t−h])×Ft−h measurable. Let

G ≡{Q ∈B ([0, t−h])×Ft−h : [h, t]×Ω∩Q+h ∈B ([0, t])×Ft}

First consider I×B where I is an interval in B ([0, t−h]) and B ∈Ft−h. Then

[h, t]×Ω∩=h+I×B

(I +h)×B = I′×B

where I′ is in B ([0, t]) and of course B ∈Ft−h ⊆Ft . Thus the sets of this form, are in G .Next suppose Q ∈ G . Is QC ∈ G ?(

[h, t]×Ω∩(QC +h

))∪ [h, t]×Ω∩ (Q+h)∪ [0,h)×Ω = [0, t]×Ω

Then all of these disjoint sets but the first are in B ([0, t])×Ft . It follows that the first isalso in B ([0, t])×Ft . It is clear that G is also closed with respect to countable disjointunions. Therefore, G contains the π system of sets of the form I×B just described. Itfollows that G = B ([0, t−h])×Ft−h.

Now if Q is progressively measurable, then [0, t−h]×Ω∩Q is B ([0, t−h])×Ft−hmeasurable and so from what was just shown, [h, t]×Ω∩Q+ h ∈B ([0, t])×Ft . ThusτhXQ is progressively measurable. It follows that if f ∈ V , you could consider φ ( f ) for