73.2. THE SITUATION 2463

73.2 The SituationNow consider the following situation. There are real separable Banach spaces V,W suchthat W is a Hilbert space and

V ⊆W, W ′ ⊆V ′

where V is dense in W . Also let B ∈L (W,W ′) satisfy

⟨Bw,w⟩ ≥ 0, ⟨Bu,v⟩= ⟨Bv,u⟩

Note that B does not need to be one to one. Also allowed is the case where B is the Rieszmap. It could also happen that V =W .

Situation 73.2.1 Let X have values in V and satisfy the following

BX (t) = BX0 +∫ t

0Y (s)ds+B

∫ t

0Z (s)dW (s) , (73.2.1)

X0 ∈ L2 (Ω;W ) and is F0 measurable, where Z is L2(Q1/2U,W

)progressively measurable

and∥Z∥L2([0,T ]×Ω,L2(Q1/2U,W)) < ∞.

This is what is needed to define the stochastic integral in the above formula. Here Q is anonnegative self adjoint operator defined on U. It could even be I.

Assume X ,Y satisfy

BX ,Y ∈ K′ ≡ Lp′ ([0,T ]×Ω;V ′),

the σ algebra of measurable sets defining K′ will be the progressively measurable sets.Here 1/p′+1/p = 1, p > 1.

Also the sense in which the equation holds is as follows. For a.e. ω, the equation holdsin V ′ for all t ∈ [0,T ]. Thus we are considering a particular representative X of K for whichthis happens. Also it is only assumed that BX (t) = B(X (t)) for a.e. t. Thus BX is thename of a function having values in V ′ for which BX (t) = B(X (t)) for a.e. t. Assumethat X is progressively measurable also and

X ∈ Lp ([0,T ]×Ω,V )

Also W (t) is a JJ∗ Wiener process on U1 in the above diagram. U1 can be assumed to beU.

The goal is to prove the following Ito formula valid for a.e. t for each ω off a set ofmeasure zero.

⟨BX (t) ,X (t)⟩= ⟨BX0,X0⟩+∫ t

0

(2⟨Y (s) ,X (s)⟩+ ⟨BZ,Z⟩L2

)ds

+∫ t

0

(Z ◦ J−1)∗BX ◦ JdW (73.2.2)