74.2. PRELIMINARY RESULTS 2499

of a function having values in V ′ for which BX (t) = B(X (t)) for a.e. t. Assume that X isprogressively measurable also and X ∈ Lp ([0,T ]×Ω,V ).

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)⟩)ds

+[R−1BM,M

](t)+2

∫ t

0⟨BX ,dM⟩ (74.1.2)

where R is the Riesz map from W to W ′. The most significant feature of the last term is thatit is a local martingale. The third term on the right is the covariation of the two martingalesR−1BM and M. It will follow from the argument that this will be nonnegative.

Note that the assumptions on M imply that [M] ∈ L1 ([0,T ]×Ω).

74.2 Preliminary ResultsHere are discussed some preliminary results which will be needed. From the integral equa-tion, if φ ∈ Lq (Ω;V ) and ψ ∈C∞

c (0,T ) for q = max(p,2) ,∫Ω

∫ T

0((BX)(t)−BM (t)−BX0)ψ

′φdtdP

=∫

Ω

∫ T

0

∫ t

0Y (s)ψ

′ (t)dsφdtdP

Then the term on the right equals∫Ω

∫ T

0

∫ T

sY (s)ψ

′ (t)dtdsφ (ω)dP =∫

Ω

(−∫ T

0Y (s)ψ (s)ds

)φ (ω)dP

It follows that, since φ is arbitrary,∫ T

0((BX)(t)−BM (t)−BX0)ψ

′ (t)dt =−∫ T

0Y (s)ψ (s)ds

in Lq′ (Ω;V ′) and so the weak time derivative of

t→ (BX)(t)−BM (t)−BX0

equals Y in Lq′([0,T ] ;Lq′ (Ω,V ′)

).Thus, by Theorem 34.2.9, for a.e. t,

B(X (t)−M (t)) = BX0 +∫ t

0Y (s)ds in Lq′ (

Ω,V ′).

That is,

(BX)(t) = BX0 +∫ t

0Y (s)ds+BM (t) , t /∈ N̂, m

(N̂)= 0